Generalized cosmological constant from gauging Maxwell-conformal algebra

The Maxwell extension of the conformal algebra is presented. With the help of gauging the Maxwell-conformal group, a conformally invariant theory of gravity is constructed. In contrast to the conventional conformally invariant actions, our gravitational action contains the Einstein-Hilbert term without introducing any additional (compensator) scalar field to satisfy the local scale invariance. This is achieved by using the curvatures of the algebra. In a special condition, we show that the resulting action is reduced to the Brans-Dicke like theory of gravity. We subsequently find the generalized Einstein field equation together with a coordinate dependent cosmological term and additional contributions.


I. INTRODUCTION
One of the essential problems of modern high-energy physics is to construct a consistent theory which contains gravity together with other interactions in a single framework.To deal with this challenge, one needs to propose a unified theory which governs the gravitational and quantum behaviours of nature.In this context, we have two different theories named as General Theory of Relativity (GR) and Quantum Field Theory (QFT).Both of them have been very successful in their conceptual frameworks with several shortcomings in extreme regimes.To unify these theories, we need to reformulate or modify them because there are some difficulties (for more detail see [1]).For this purpose, the symmetry feature of nature may play a significant role in the mentioned problem because symmetries simplify the mathematical complexity of the physical systems [2].One of the best examples is the standard model in which local symmetry plays the predominant role.All of the four fundamental interactions in nature, except gravity, are described by this model.This theory based on the global gauge symmetry principle.Furthermore, motivating from Yang and Mills [3], understanding the gravitational interaction as a gauge theory started with Utiyama's work [4].He proposed that GR can be obtained as a gauge theory based on the local homogeneous Lorentz group.From this idea, Kibble and Sciama constructed a gauge theory of gravity which contains GR with non-zero torsion based on localization of the Poincare group [5,6].After these significant works, many space-time groups were analyzed in the gauge theoretical context such as the Weyl [7][8][9][10][11][12], affine [13][14][15] and conformal [16][17][18][19][20][21][22][23][24][25] groups.These studies lead to gravitational theories more general than GR.This idea is important because considering gravity as a theory based on local space-time symmetries corresponds to a considerable advance towards the unified theory [26].
It is known that the conformal transformations play an important role in physics.For instance, in order to deal with the unification problem, the conformal gauge theory of gravity is seen a good candidate [27][28][29][30][31][32] since it is expected to be renormalizable.Furthermore, there are several uses of the conformal group C (1, 3) such as the quantum field theory in the high energy regime, cosmological constant problem [31], dark matter and dark energy [33,34].For these reasons, C (1, 3) has reached much interest and extensive study in the literature.In the gravitational sector, the conformal gravity can be seen as a natural extension of GR with additional symmetries [35][36][37][38] but it has some restrictions for the construction an invariant action.Invariance of the gravitational action for the local conformal transformations can be constructed by the terms either linear or quadratic in curvature forms.In curvature-linear actions, we need to introduce a Brans-Dicke like compensating scalar field to satisfy the scale invariance [39,40].This approach leads to generalizing Einstein's field equations and it can be reduced to the standard GR in a special condition.On the other side, if we use a curvature-quadratic action that can be constructed by the square of the Weyl tensor, we get higher-order theory of gravity which does not contain Einstein's gravity (more detail see [41]).
To avoid the mentioned restrictions on GR, we consider extending the conformal group with the Maxwell symmetry.The Maxwell group M (1, 3) is a symmetry extension which contains non-commutative momentum operator as [P a , P b ] = iZ ab [42][43][44][45][46]. Here, the antisymmetric central charges Z ab represent additional degrees of freedom.
According to the early studies [47,48], the Maxwell symmetry has been assumed to describe a particle moving in a Minkowski space-time filled with a constant electromagnetic background field.In recent years the Maxwell group and its modifications have gained much interest in the context of producing an alternative way to generalize Einstein's theory of gravity and supergravity [49][50][51][52][53][54][55][56][57][58][59][60][61][62].In particular, gauging the Maxwell (super)algebra provide an effective background to obtain generalized cosmological constant term [49-51, 53, 56, 57, 60].In this extended framework, the new degrees of freedom represent uniform gauge field strengths in (super)space which leading to uniform constant energy density [45].Therefore, we can say that the Maxwell symmetry allows a powerful framework to generalize GR.Besides, this symmetry can be used in various areas such as describing planar dynamics of the Landau problem [63], the higher spin fields [64,65], and it was also used in the string theory as an internal symmetry of the matter gauge fields [66].
Our main motivation on this extension is to develop a theory respecting the local conformal symmetry.For this, we imposed certain constraints in order to break the initial symmetry.After the symmetry breaking, the action in the present work yields the generalized Brans-Dicke theory when the Maxwell-conformalMC (1, 3) symmetry is broken down to the conformal group.
The organization of this letter is as follows.In Section II, we briefly recall the conformal algebra c (1, 3) and we then obtain the Maxwell-Conformal algebra mc (1, 3) by taking account of the isomorphism between C (1, 3) and SO (2, 4) groups.In Section III, the gauge theory of MC (1, 3) group is constructed.In Section IV, after a brief discussion of the conformal gravity, we establish a new conformal invariant action that contains the Einstein-Hilbert term together with the contributions which come from the Maxwell symmetry.We show that the action can be reduced to a Brans-Dicke like theory of gravity by considering some constraints.We also give a generalized version of the Einstein field equations together with a dynamical cosmological term and an additional energy-momentum tensors.In the last section, we conclude our work with some comments and possible future developments.

II. THE CONFORMAL ALGEBRA AND ITS MAXWELL EXTENSION
In this section, we consider the conformal group C (1, 3) and its Maxwell extension in four-dimensional space-time.At first, we give a brief summary of C (1, 3).The conformal group was introduced into physics by Bateman and Cunningham [67][68][69] in the framework of the electromagnetic theory.It can also be interpreted as the largest group of coordinate transformation which preserves the light-cone structure [70].This symmetry contains the Lorentz, translations, special conformal transformations, and dilation transformations.Furthermore, the Lie groups of C (1, 3) and SO (2, 4) are locally isomorphic, so they satisfy the same algebra.Since the generators J AB of SO (2, 4) group satisfy the following commutation relations, where the metric η AB = diag (+, −, −, −, −, +) and the capital Latin indices run A, B, ... = 0, 1, 2, 3, 5, 6.The generators of conformal transformations are represented by X A = {P a , K a , D, M ab } which correspond to the translations, special conformal transformations dilatations and Lorentz transformations can be realized through J AB generators as follows [71,72], where l is the unit of length (L).Thus, the commutation relations for the generators in Eq.( 2) can be written as follows, Therefore, we get the usual conformal Lie algebra c (1, 3).Here, η ab = diag (+, −, −, −) and the small Latin indices have the range a, b, ... = 0, 1, 2, 3. Analogously to the approach taken above, one can extend SO (2, 4) algebra in Eq.( 1) by an additional antisymmetric central charges Z AB as follows [50,54], where J AB correspond to a generalized SO (2, 4) generators.The resulting extended algebra can be named as the Maxwell extension of so (2, 4) algebra (mso (2,4)).In order to find the Maxwell extension of c (1, 3) algebra, decomposing J AB and Z AB into the following generators, and checking all possible commutators of the generators, the Lie algebra of the corresponding group is found to be while the remaining commutators are zero.Here, the constant λ have the unit of L −2 which will be considered as the cosmological constant.This algebra can be called as the Maxwell-conformal algebra mc (1, 3).Here, the additional generators Z ab , Z a , Za and Z represent tensor, vector and scalar contributions of the Maxwell symmetry, respectively.The self-consistency of this algebra can be checked by the help of the Jacobi identities.We also note that the decomposition of the higher dimensional tensor extended Maxwell algebras provide a powerful framework to generate various forms of the Maxwell algebras.For example, if we choose the generators as and assume a special condition as Z 5a = −Z 6a , we get the minimal Maxwell algebra [42,49] as follows, Moreover, it is easy to see that these algebras are reduced to their standard forms when taking Z AB = 0.

III. GAUGING THE MAXWELL-CONFORMAL ALGEBRA
In the previous section, we have obtained the Maxwell extension of the conformal algebra in Eq.( 6).This may lead to interesting results because we know that the conformal transformations play an important role in various areas of physics.In particular, the conformal transformations provide a useful framework to develop generalized theories of gravity.In this context, we will establish a gravitational gauge theory based on the Maxwell-conformal group.So we start with mc (1, 3) algebra that acts on a four-dimensional space-time.We will follow the similar methods for constructing a gauge theory of gravity as in [3-5, 17, 23] by using differential forms [49,53].Let us first define the Maxwell-conformal algebra-valued one-form gauge field A (x) = A A (x) X A as follows, where A A (x) = e a , c a , χ, ω ab , B ab , r a , ra , r are the gauge fields which correspond to the generators X A = P a , K a , D, M ab , Z ab , Z a , Za , Z , respectively.Besides, the unit dimension of gauge fields are considered as [e a ] = L, and the remaining gauge fields are dimensionless.The variation of the gauge field A (x) under a gauge transformation can be found by using the following formula, where ζ (x) represents a MC (1, 3)-valued zero-form gauge generator which is defined as follows, where y a (x), n a (x), ρ(x), τ ab (x) φ ab (x), s a (x), sa (x) and s (x), are the parameters of the corresponding generators.Thus, by the help of Eqs.( 6) and (10), one can find the transformation law of gauge fields as follows, The curvature two-forms of the associated gauge fields F (x) = F A (x) X A are defined to be where and R (Z) represent the curvatures which comes from the associated generators.In order to find the explicit forms of these curvatures, we use the following structure equation, and taking account of the gauge fields in Eq.( 9), the group curvatures are found to be, where denotes the usual Riemann curvature tensor.We note that the curvatures R (P ) a , R (K) a , R (D) and R (M ) ab have the same form as in the standard conformal gravity.Moreover, the new curvature R (Z) ab contains λ-dependent term that recalls the contribution to the Riemann curvature tensor as in (A) dS gravity.This is the well-known characterstics of the Maxwell-like algebras.The transformation properties of the two-form curvatures under the infinitesimal gauge transformation can be found by the following expression and Eq.( 11), and thus one can obtain, Taking the exterior covariant derivative of the curvatures, we can find the Bianchi identities as where D is the Lorentz-Weyl covariant derivative which is defined as Here, ω is the spin connection and w is the conformal weight of the corresponding field.

IV. GRAVITATIONAL ACTION FOR MC (4)
We introduce our discussion with a summary of the previous treatments of the conformal gravity.The conformal transformation of the metric tensor g µν (x) can be given as follows, where Ω (x) is an arbitrary, positive and dimensionless space-time function which is known as the conformal factor.The transformations in Eq.( 20) preserve the light-cone structure and the angles between vectors [26].This feature restricts to construct an invariant action under conformal transformations.For instance, the well-known Einstein-Hilbert (EH) action does not satisfy the local conformal invariance.In the literature, there are several methods to overcome this problem but two of these have attracted much attention.The first one is to construct an action linear in curvature scalar together with a Brans-Dicke like [39] scalar compensating field φ (x) having conformal weight (−1).
In the context of this method, Deser [73] and Dirac [40] used a conformal invariant φ 2 R term rather than R in the EH action to satisfy local conformal invariance, By the help of this approach, the generalized versions of Einstein's field equations were obtained [8][9][10][11][12].The second one is to use a curvature-quadratic action without a compensating field in four dimensions as follows, where C µνρσ is the Weyl tensor and α is a coupling constant.This theory is also known as "Weyl-squared" theory.This type of action does not provide an exact relation with Einstein's gravity but it leads to higher-order field equations [41].
Employing quadratic gravity where terms quadratic in the curvature tensor, a Weyl like (super)gravity is obtained as a gauge theory of the (super)conformal group in [16,17].
In this section, we want to construct a conformal invariant action which contains EH term together with additional contributions.For this purpose, starting from the invariance principle, our research focuses primarily on a free gravitational theory based on the gauge theory of the Maxwell-conformal group.We initially define a shifted curvature having zero conformal weight as, where µ is an arbitrary dimensionless constant.Taking account of the Eq.( 18) and considering the constraints , it is easy to see that the gauge transformation of the shifted curvature can be found as δR (J) . By the help of this definition, we can write the Euler or Gauss-Bonnet like topological gravitational action as follows, where κ = 8πGc −4 is Einstein's gravitational constant and * denoting the Hodge star dual operator.If we take the constant µ = 4 and ignoring the total derivative term, the action becomes The first two terms correspond the Einstein-Hilbert action together with a cosmological term, and the remaining terms contain the higher-order curvature term and mixed terms with the new extra fields B ab (x), r a (x) and ra (x) coupled to the spin connection, vielbein and special conformal gauge field.Also, the cosmological constant appeared in the action as linearly and inversely.Similar formulation was analyzed in [74] as a dynamical structure of quantum cosmological constant.The resulting action is conformal invariant but not invariant under the local transformations of MC (1, 3) group.This means that the Maxwell parts of MC (1, 3) symmetry group were broken.In this construction, the Maxwell fields B ab (x), r a (x) and ra (x) were employed to satisfy the local conformal invariance of the action.Therefore, we obtain an action that provide a useful framework to construct various gravitational theories because it includes additional tensor, vector and scalar fields which come from the Maxwell symmetry.For instance, if both tensors R(P ) a and R(K) a are set simultaneously equal to zero, then the constraints of the theory yield that their corresponding gauge fields e a (x) and c a (x) are related by [75] where ϕ (x) is a Brans-Dicke like dimensionless scalar compensating field having the following transformation, Naturally, we can expect another relationship between gauge parameters as in Eq.( 26), Using these relationships, the action in Eq.( 25) takes the following form, Similarly, if we impose the following relation, ra + λϕr a := 0, (30) the action reduces to a generalized version of the Brans-Dicke theory of gravity, This action coincides with the result given by Chamseddine [76] when the Maxwell extension suppressed and is a generalization of the action given in [53].Now we are in a position to consider the field equations of the theory which can be derived from a variational action principle.Taking into account of the Eq. ( 12), the equations of motion can be found by the variation of the action in Eq.( 24) with respect to the gauge fields ω ab (x) , e a (x) ,B ab (x) and ϕ (x) respectively, Furthermore, one can show that all these equations of motion verify each other.If we consider a special solution R (M ) ab = −4R (Z) ab , the equations of motion are satisfied.Making use of the shifted curvature and Eq.( 33), we get the following equation, Passing from the tangent indices to world space-time indices [49,55] as, then considering Eq.( 15) and Eq.( 16), one gets the following field equation, where, here the term 6λ 1 + ϕ 2 corresponds a generalized cosmological term and T µ ρ (B) represents the tensorial contribution of the Maxwell symmetry.Therefore, we demonstrated that the new extended framework leads to the generalized Einstein field equation together with a coordinate dependent cosmological term plus additional energy-momentum tensors.Moreover, in the limit of B (x) = 0 and ϕ (x) = 0, we get the well-known gravitational field equation with the cosmological constant, V. CONCLUSION In four-dimensional space-time, as we mentioned before, there are several restrictions on the construction of the conformally invariant Einstein Hilbert action.To overcome these restrictions, according to the classical conformal gauge theory, it is required to introduce either a scalar compensating field which has a certain conformal weight [39,40,73] or modify the related gauge symmetry [24,41].Furthermore, it is known from the literature that the cosmological constant problem may require an alternative approach to gravity.In the present paper, we wanted to find a locally conformal invariant theory of gravity especially for two reasons.The first, we know that the conformal invariance is very significant field in high-energy physics.According to the literature, it was thought that the gauge theory of conformal group is an important study area to resolve the quantum theory gravity and the unification problems [27][28][29][30][31][32].The second, to cope with the difficulties to establishing conformal invariant gravity theories, we looked for to construct an alternative way to obtain a gravitational action including the Einstein-Hilbert term which is invariant under local conformal transformations by using an extended gauge group.From these motivations, we have found a Maxwell extension of the conformal group in Eq.( 6).This is done by considering the isomorphism between SO(2, 4) and C(1, 3).We then established a gauge theory of gravity based on the Maxwell-conformal group.We found a local conformally invariant gravitational action by breaking Maxwell part of related symmetry group in Eq. (25).The resulting gravitational action contains the Einstein-Hilbert term together with a cosmological constant and additional source terms.This action was constructed by using the shifted curvature two-forms in Eq.( 23) which was constituted by the curvatures of the MC (1,3).
In this theory, there are additional tensor, vector, and scalar fields in comparison with the conformally invariant Brans-Dicke theory gravity.These properties of the theory provide more general mathematical framework to obtain different kind of gravity theories.Using these tools, in a special condition given in Eq.( 26) and Eq.( 30), this extended gravitational theory was reduced to a generalization of Brans-Dicke theory of gravity [70,77] in Eq. (31).Subsequently, we obtained a new generalization of Einstein's field equations which contain a dynamical cosmological term and additional energy-momentum tensor Eq. (38).We also obtained the standard formulation of Einstein's field equation with a cosmological constant in the limit of B ab (x) = 0 and ϕ (x) = 0 in Eq. (40).
In conclusion, we have presented a new conformally invariant geometric structure for gravitational theory including a generalized cosmological constant in the framework of the Maxwell-conformal algebra.Therefore the gravitational theory constructed here may play a role in the resolution of the mentioned problems.On the other hand, if one wants to construct an invariant action under local MC (1, 3) symmetry, the Stelle-West like method [25,75,78] provides a useful background.According to this method, one can construct an action in six-dimensional space and making dimensional reduction to four-dimensional we hope that it is possible to find an invariant action under the local MC (1, 3) transformations.We are working on this approach.
We also note that decomposition of higher dimensional algebras may allow a powerful method to obtain various types of the Maxwell algebras.In the spirit of this approach, analogously to the construction of MC (1, 3), our studies on this subject are in progress.
+τ a b r b + ρr a − 2φ a b e b + se a − 1 2λ ρc a , δr a = −ds a − ω a b sb + χs a + 2B a b n b + rn a − λ 2 χy a , +τ a b rb − ρr a − 2φ a b c b − sc a + λ 2 ρe a , δr = −ds − y a c a − 2y a ra + 2n a r a + n a e a − 2s a c a + 2s a e a .