5 April 2001
Physics Letters B 504 (2001) 33–37 www.elsevier.nl/locate/npe
Deforming Einstein’s gravity Ali H. Chamseddine Center for Advanced Mathematical Sciences (CAMS), and Physics Department, American University of Beirut, Lebanon Received 1 November 2000; accepted 21 November 2000 Editor: L. Alvarez-Gaumé
Abstract A deformation of Einstein gravity is constructed based on gauging the noncommutative ISO(3, 1) group using the Seiberg– Witten map. The transformation of the star product under diffeomorphism is given, and the action is determined to second order in the deformation parameter. 2001 Published by Elsevier Science B.V.
1. Introduction Open string theories as well as D-branes in the presence of background antisymmetric B-field give rise to noncommutative effective field theories [1–7]. This is equivalent to field theories deformed with the star product [8,9]. The primary example of this is noncommutative U (N) Yang–Mills theory [1,7]. In recent works [10,11], it was argued that the gravitational field gets deformed and becomes complex [12]. The hermitian metric [13], includes both the symmetric metric and an antisymmetric tensor. The analysis of the linearized theory showed that the theory is consistent, however, further work is needed to show that this is maintained at the nonlinear level, a basic problem faced by all theories of nonsymmetric gravity [14–16]. For this to happen, it is essential that the diffeomorphism invariance of the real theory is generalized to the complex case. This can happen when both the diffeomorphism transformations and the abelian gauge transformations of the antisymmetric tensor combine to form complex diffeomorphism.
E-mail address:
[email protected] (A.H. Chamseddine).
The need for this is that gauge symmetry prevents the ghost degrees of freedom present in the antisymmetric tensor from propagating. The main argument for considering complex vielbein and gauged U (1, D − 1) is that for noncommutative Yang–Mills theory it is only possible to gauge the U (N) Lie algebras [7]. Reality conditions necessary to consider SO(N) or SP(N) Lie algebras are not possible. This obstacle was overcome [10,11], by realizing that it is possible to define subgroups of orthogonal and symplectic subalgebras of noncommutative unitary gauge transformations even though the gauge fields are not valued in the subalgebras of the U (N) Lie algebra. What makes this possible is that one can generalize the reality condition to act on the deformation parameter. Thus the gauge fields are taken to be functions of the deformation parameters θ and the expansion in terms of the nondeformed fields is given by the Seiberg–Witten map. To construct a noncommutative gravitational action in four dimensions one proceeds as follows. First the gauge field strength of the noncommutative gauge group SO(4, 1) is taken. This is followed by an Inonü–Wigner contraction to the group ISO(3, 1), thus determining the dependence of the deformed vierbein on the undeformed one. At this
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stage the construction of the deformed curvature scalar becomes straightforward. The deformed action is computed up to second order in θ . The plan of this Letter is as follows. In Section 2 we review the conditions allowing to deal with noncommutative SO(N) algebras, derive the noncommutative gauge field strengths, perform the group contraction and give the deformed curvature scalar. In Section 3 we expand the action to second order in θ . Section 4 is the conclusion.
2. Noncommutative gauging of SO(4, 1) The U (N) gauge fields are subject to the condition Aˆ †µ = −Aˆ µ . Such condition can be maintained under the gauge transformations Aˆ gµ = gˆ ∗ Aˆ µ ∗ gˆ∗−1 − gˆ ∗ ∂µ gˆ∗−1 , ˆ Therefore, we first inwhere gˆ ∗ gˆ∗−1 = 1 = gˆ∗−1 ∗ g. troduce the gauge fields ωˆ µAB , subject to the conditions [10,11] ωˆ µAB† (x, θ ) = −ωˆ µBA (x, θ ), ωˆ µAB (x, θ )r ≡ ωˆ µAB (x, −θ ) = −ωˆ µBA (x, θ ). Expanding the gauge fields in powers of θ , we have
To solve this equation we first write ωˆ µAB = ωµAB + ωµAB (ω), λˆ AB = λAB + λAB (λ, ω), where ωµAB (ω) and λAB (λ, ω) are functions of θ , and then substitute into the variational equation to get [7] ωµAB (ω + δω) − ωµAB (ω) = ∂µ λAB + ωµAC λCB − λAC ωµCB + ωµAC λCB i − λAC ωµCB + θ νρ ∂ν ωµAB ∂ρ λCB 2 + ∂ρ λAC ∂ν ωµCB . This equation is solved, to first order in θ , by AB i ωˆ µAB = ωµAB − θ νρ ων , ∂ρ ωµ + Rρµ + O θ2 , 4 i ˆλAB = λAB + θ νρ {∂ν λ, ωρ }AB + O θ 2 , 4 where we have defined the anticommutator {α, β}AB ≡ α AC β CB + β AC α CB . With this it is possible to derive the differential equation that govern the dependence of the deformed fields on θ to all orders i ρµ AB , δ ωˆ µAB (θ ) = − θ νρ ωˆ ν ,∗ ∂ρ ωˆ µ + R 4 with the products in the anticommutator given by the star product, and where
AB + ··· . ωˆ µAB (x, θ ) = ωµAB − iθ νρ ωµνρ
AB µν R = ∂µ ωˆ νAB − ∂ν ωˆ µAB + ωˆ µAC ∗ ωˆ νCB − ωˆ νAC ∗ ωˆ µCB .
The above conditions then imply the following
We are mainly interested in determining ωˆ µAB (θ ) to second order in θ . This is due to the fact that the deformed gravitational action is required to be hermitian. The undeformed fields being real, then implies that all odd powers of θ in the action must vanish. The above equation could be solved iteratively, by inserting the solution to first order in θ in the differential equation and integrating it. The second order corrections in θ to ωˆ µAB are
ωµAB = −ωµBA ,
AB BA ωµνρ = ωµνρ .
A basic assumption to be made is that there are no new degrees of freedom introduced by the new fields, and that they are related to the undeformed fields by the Seiberg–Witten map [7]. This is defined by the property ωˆ µAB (ω) + δλˆ ωˆ µAB (ω) = ωˆ µAB (ω + δλ ω), ˆ
where gˆ = eλ and the infinitesimal transformation of ωµAB is given by δλ ωµAB = ∂µ λAB + ωµAC λCB − λAC ωµCB , and for the deformed field it is δλˆ ωˆ µAB = ∂µ λˆ AB + ωˆ µAC ∗ λˆ CB − λˆ AC ∗ ωˆ µCB .
1 νρ κσ θ θ 32 ×
ωκ , 2{Rσ ν , Rµρ } − ων , (Dρ Rσ µ + ∂ρ Rσ µ ) AB − ∂σ ων, (∂ρ ωµ + Rρµ )
AB + ∂ν ωκ , ∂ρ (∂σ ωµ + Rσ µ ) AB − ων , (∂ρ ωκ + Rρκ ) , (∂σ ωµ + Rσ µ ) .
A.H. Chamseddine / Physics Letters B 504 (2001) 33–37
One problem remains of how to determine the dependence of the vierbein eˆµa on the undeformed field as it is not a gauge field. To resolve this problem we adopt the strategy of considering the field eµa as the gauge field of the translational generator of the inhomogeneous Lorentz group, obtained through the contraction of the group SO(4, 1) to ISO(3, 1). This is done as follows. First define the SO(4, 1) gauge field ωµAB with the field strength AB = ∂µ ωνAB − ∂ν ωµAB + ωµAC ωνCB − ωνAC ωµCB , Rµν
and let A = a, 5. Define ωµa5 = keµa . Then we have ab = ∂µ ωνab − ∂ν ωµab + ωµac ωνcb − ωνac ωcb Rµν + k 2 eµa eνb − eνa eµb , a5 a Rµν ≡ kTµν = k(∂µ eνa − ∂ν eµa + ωµac eνc − ωνac eµc ).
The contraction is done by taking the limit k → 0. By a = 0 one can solve for ωab imposing the condition Tµν µ a in terms of eµ . We shall adopt a similar strategy in the deformed case. We write ωˆ µa5 = k eˆµa and ωˆ µ55 = k φˆ µ . a = 0 and not We shall only impose the condition Tµν a = 0 because we are not interested in φ which Tˆµν µ will drop out in the limit k → 0. The result for eˆµa in the limit k → 0 is i ac c eˆµa = eµa − θ νρ ωνac ∂ρ eµc + ∂ρ ωµac + Rρµ eν 4 1 + θ νρ θ κσ 2{Rσ ν , Rµρ }ac eκc 32 − ωκac Dρ Rσcdµ + ∂ρ Rσcdµ eνd ad d − ων , Dρ Rσ µ + ∂ρ Rσ µ eκ ac c − ∂σ ων , (∂ρ ωµ + Rρµ ) eκ cd d eν − ωκac ∂σ ωνcd ∂ρ eµd + ∂ρ ωµcd + Rρµ c ac c ac ac + ∂ν ωκ ∂ρ ∂σ eµ − ∂ρ ∂σ ωµ + Rσ µ ∂ν eκ ac − ων , (∂ρ ωκ + Rρκ ) ∂σ eµc − ∂σ ωµac + Rσacµ cd d eν × ωνcd ∂ρ eκd + ∂ρ ωκcd + Rρκ 3 +O θ . At this point, it is possible to determine the deformed curvature and use it to calculate the deformed
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action given by √ ν † √ † µ ab µν ∗ e∗b ∗ eˆ . d 4 x eˆ ∗ eˆ∗a ∗ R Notice that this action is hermitian including the measure. We have defined eˆ = det(eˆµa ), and the inverse vierbein by µ
eˆ∗a ∗ eˆµb = δab . This will determine the inverse deformed vierbein as an expansion in θ . By writing a a + θ νρ θ κσ eµνρκσ + O θ3 , eˆµa = eµa + iθ νρ eµνρ µ µ µ eˆ∗a = eaµ + iθ νρ eaνρ + θ νρ θ κσ eaνρκσ + O θ3 , a a and eµνρκσ can be read from the expansion where eµνρ a of eˆµ to second order in θ , one finds that µ µ b eaνρ = −eb eaκ eκνρ + 12 ∂ν eaκ ∂ρ eκb , µ α µ b b eaνρκσ = −eb eaνρ eακσ + eaα eανρκσ
− 14 ∂ν ∂κ eaα ∂ρ ∂σ eαb
b α . − 12 ∂ν eaα ∂ρ eακσ + 12 ∂ρ eαb ∂ν eaκσ It is legitimate to question the meaning of the star product under general coordinate transformations, and whether this action is diffeomorphism invariant. Afterall, the original definition of the star product assumes that the commutator [x µ , x ν ] = iθ µν is constant. However, under diffeomorphism transformations, θ µν becomes a function of x, and one has to generalize the definition of the star product to be applicable for a general manifold. The prescription for doing this was given by Kontsevich [17]. The star product is then defined by 2 f ∗ g = fg + hB ¯ 1 (f, g) + h¯ B2 (f, g) + · · · ab ∂a f ∂b g = fg + hα ¯ 1 + h¯ 2 α ab α cd ∂a ∂c f ∂b ∂d g 2 1 + h¯ 2 α as ∂s α bc (∂a ∂b fc ∂g + ∂a ∂b g∂c f ) 3 + O h¯ 3 ,
where α ab = θ µν ∂µ za ∂ν zb + O θ 3 .
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Therefore, under diffeomorphisms the star product transforms according to ∗ → ∗ , f (h¯ ) = Df (h¯ ), f ∗ g = D D −1 f ∗ D −1 g ,
D=1+ h¯ Di . i=1
In this case [18] h¯ 2 µν ρσ θ θ ∂µ ∂ρ za ∂ν ∂σ zb ∂a ∂b 4 + 23 ∂µ ∂ρ za ∂ν zb ∂σ zc ∂a ∂b ∂c + O h¯ 4 .
D=1−
It is thus possible to define the star product to be invariant under diffeomorphism transformations.
3. The action to second order in θ
where ab ab ac ac = ∂µ ωνρτ + ωµac ωνρτ + ωµρτ ωνcb Rµνρτ 1 − ∂ρ ωµac ∂τ ωυcb − µ ↔ ν, 2 ab ab ac ac ac cb Rµνρτ κσ = ∂µ ωνρτ κσ + ωµ ωνρτ κσ + ωµρτ κσ ων 1 ac ac − ωµρτ ωνκσ − ∂ρ ∂κ ωµac ∂τ ∂σ ωυcb 4 − µ ↔ ν.
With this we can now expand ν † µ ab µν eˆ∗a ∗ R ∗ e∗b ab ν µ ab ν = R + θ ρτ θ κσ eaµ Rµνρτ κσ eb + eaρτ κσ Rµν eb
+ O θ4 .
µ ab ab ν − eaρτ Rµνκσ ebν − eaµ Rµνρτ ebκσ
det eˆµa a = 1 + iθ ρτ eaµ eµρτ a a µ ν 1 a b + θ ρτ θ κσ eµρτ eνκσ ec ec − eµρτ eνκσ eaµ ebν 2 + O θ3 . This completes the action to second order in θ . Of course the actual expression obtained after substitutµ µ ab and ωab ing for the fields eaρτ , eaρτ κσ , ωµρτ µρτ κσ is very complicated, and it is not clear whether one can associate a geometric structure with it. One can, however, take this expression and study the deformations to the graviton propagator, which will receive θ 2 corrections. It is an interesting question to study the effect of the deformation on the renormalizability of the theory. This project will have to be handled by using a computer program for algebraic manipulations.
4. Conclusion
To determine the action to second order in θ , we first write 3 ab ab ab ab µν = Rµν + iθ ρτ Rµνρτ + θ ρτ θ κσ Rµνρτ R κσ + O θ ,
ab ν µ ab ν + eaµ Rµν ebρτ κσ − eaρτ Rµν ebκσ
is
Notice that the odd powers of θ cancel because of the hermiticity of the above expression and reality of the undeformed fields. The expansion of the determinant
In this work we have shown that it is possible to deform Einstein’s gravity without introducing new fields. The idea is based on the gauging of the noncommutative gauge group ISO(3, 1) and using the Seiberg– Witten map to express the deformed fields in terms of the undeformed ones. The reality of the undeformed fields and the hermiticity of the action implies that the lowest order correction to Einstein’s action is second order in the deformation parameter. This makes the form of the corrections fairly complicated. It is, however, possible to use perturbation theory to determine the modified graviton propagator as well as the vertex operators, to second order in θ . It is an interesting problem to study the renormalizability of the theory and the effects of the deformation parameter on the infrared and ultraviolet divergencies. Performing these calculations will be left for future work.
Acknowledgements I would like to thank the Alexander von Humboldt Foundation for support through a research award. I would also like to thank Slava Mukhanov for hospitality at the Ludwig-Maxmiilans University in Münich where part of this work was done.
A.H. Chamseddine / Physics Letters B 504 (2001) 33–37
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