Weyl-invariant extension of the Metric-Affine Gravity

Metric-affine geometry provides a non-trivial extension of the general relativity where the metric and connection are treated as the two independent fundamental quantities in constructing the space-time (with non-vanishing torsion and non-metricity). In this paper we study the generic form of action in this formalism, and then construct the Weyl-invariant version of this theory. It is shown that in Weitzenbock space, the obtained Weyl-invariant action can cover the conformally invariant teleparallel action. Finally the related field equations are obtained in the general case.


Introduction
Extended theories of gravity have become a field of interest in recent years due to the lack of a theory which could fully describe gravity as one of the fundamental interactions together with other ones.
Shortcoming of the current gravitation theory in cosmological scales, despite all the successes in solar system tests, is another reason in developing the idea of extending it and has confirmed the need to surpass Einstein's general theory of relativity (GR) [1,2]. The extension can be made in different ways such as geometrical, dynamical and higher dimensional or even a combination of them [3,4]. In this paper, we will focus on geometric extension to gravity under the notion of metric-affine formalism.
Our current description of gravity is based, via theory of GR, on Riemannian geometry in which metric is the only geometrical object needed to determine the space-time structure and connection is considered to be metric-connection, the well known Christoffel symbols. In such a context, as a priori, the connection is symmetric i.e. torsion-free condition and is also assumed to be compatible with the metric (∇ λ g µν = 0). To enlarge this scheme one can set the two presumptions aside and think of metricaffine geometry in which metric and connection are independent geometrical quantities. Therefore, connection is no longer compatible with the metric and torsion-free condition is relaxed, thus in addition to Christoffel symbols, the affine connection would contain an anti-symmetric part and non-metric terms as well. The transition from Riemann to metric-affine geometry has led to Metric-Affine Gravity (MAG), an extended theory of gravity which has been extensively studied during recent decades from different viewpoints and a variety of non-Riemannian cosmology models are proposed based on it (for example see [5]).
The underlying idea of GR that explains gravity in terms of geometric properties of space-time remains unaffected in MAG and just new concepts are added to this picture by introducing the torsion and non-metricity tensors in description of gravity. Almost all of the space-time geometries such as Riemann-Cartan, Weyl, Minkowski, etc. can be obtained by constraining the three aforementioned tensors [6,7]; this is one of the peculiarities of metric-affine formalism.
In this paper we study the conformally invariant metric-affine theory. The conformal theories of gravity are in great importance, for example, in the framework of quantum gravity it is proved that such theories are better to re-normalize [8] and also from the phenomenological point of view it is shown that the solutions of conformal invariant theories (e.g. Weyl gravity) can explain the extraordinary speed of rotation curves of galaxies and also they can address the cosmological constant problem [9]. Thus considering the conformal invariance within the context of MAG seems to be worthwhile which has been studied in some papers [10,11]. We first consider the action in its general format in MAG and then fix the degrees of freedom by imposing the conformal invariance.
The organization of the paper is as follows: the conventions and general aspects of metric-affine geometry are briefly reviewed in section 2. In section 3, the most general second order action in metric-affine formalism is presented in terms of free parameters. Weyl-invariant version of the action is investigated in section 4 by determining the free parameters and the subject will be concluded in the fifth section. The field equations of the general action are presented in appendix.

Metric-affine geometry
As mentioned the main idea in MAG is the independence of metric and connection, both of which being the fundamental quantities indicating the space-time structure and carry their own dynamics in contrast with GR where metric is the only independent dynamical variable [12,13]. Note that in Palatini formalism of GR such an assumption holds, but it brings nothing novel to the theory. In presence of the gravitational field, space-time can be curved or twisted and these features of space-time are characterized by curvature and torsion tensors defined in terms of the affine connection and its derivatives as These two definitions are obtained from the covariant derivatives commutator acting on a vector field, where ∇ denotes covariant derivative associated with the affine connection and for a tensor field is defined It must be emphasized that the third index of the connection is conventionally chosen to be in the direction along which differentiation is done 2 [12,14].
Non-metricity tensor is the other geometrical object defined as from which it is implied that inner product of vectors and so their length and the angle between them are not conserved when parallel transported along a curve in space-time [3,15].
Another difference arisen in this set-up is the increase of dynamical degrees of freedom. Noting that the affine connection is independent of the metric and is not constrained in general, thus it contains 64 independent components in four-dimensional space-time. As it is obvious by definitions given above, torsion and non-metricity are rank-three tensors being anti-symmetric in first and symmetric in last pair of indices respectively. Due to these symmetry properties the two quantities will constitute the 64 independent components of the affine connection (24 components of torsion tensor and 40 components of non-metriciy tensor). Thereby the overall number of the independent components will become 74 by taking into account the 10 independent components of the metric -a symmetric second rank tensortogether with the connection.
It is straightforward to show that the affine connection can be expressed in the following form where λ µν is the usual Christoffel symbol and the rest is a combination of torsion and non-metricity tensors defined as follows Accordingly, the affine curvature tensor will take the form of where R λ βµν ({}) indicates Riemann curvature tensor and braces are used to show covariant derivative in Riemann geometry. Contraction of (8) will result in two different rank-two tensors [3,12]; one of them is the affine Ricci tensor obtained by contracting the first index with the last pair of indices (R βν ≡ R λ βλν ) and the other one is the so-called homothetic curvature tensor which is resulted from contraction of the first two indices (R µν ≡R λ λµν ). Generation of the two second rank curvature tensors is due to the fact that the only symmetry of the affine curvature tensor is the anti-symmetric property of the last pair of indices. However, there is just one independent affine Ricci scalar and the contraction of homothetic curvature tensor with metric gives rise to vanishing scalar because of its anti-symmetric nature [3,12]. Having more complete description of geometry, in the following section we will focus on constructing the generic gravitational action based on metric-affine geometry.

Metric-affine second order action
With independence of the metric and connection in MAG, the general gravitational action is expressed in terms of metric, connection and their derivatives. Also the coupling of matter action with connection in addition to metric and matter fields is allowed in the case of MAG which forms the dissimilarity between this approach and the Palatini [12,16]. In this paper we limit our discussion to the geometrical part of the action, but what the form of the action is. Obviously, the first choice is to replace the Einstein-Hilbert (EH) action with its counterpart in metric-affine formalism, but this is not the only possibility and extension of EH action into a more general one through the metric-affine formalism can be done by considering torsion and non-metricity tensors as well as curvature tensor in construction of the action [17]. In order to construct the gravitational Lagrangian, we follow the approach of [12] and [20] which is an effective field theory approach and appropriate scalar terms are constructed at each order by applying power counting analysis. To start with and in natural units c = ℏ = 1, by choosing [dx] = [dt] = [l] all of the geometrical quantities which are needed in construction of the action, are expressed in terms of length dimension and, consequently, to make the action dimensionless, the coupling constant is related to Planck length l p . The highest power of the length dimension in the scalars shows the order of action so the first term of the generalized action which is the Ricci scalar that is replaced by the affine one, is of second order.
Restricting our discussion to second order action and considering symmetries of torsion and nonmetricity tensors, there will be four scalars written in terms of torsion tensor and its derivative 3 , eight terms made up of the non-metricity tensor and its derivative and three terms constructed from the contraction of torsion and non-metricity tensors. Accordingly, the generic second order gravitational action in n dimensions takes the following form 4 in which S µ ≡ S µβ β and a i 's denote different coupling constants, g stands for the determinant of metric, and κ contains the Planck length, e.g. in four dimensions κ = 16πl 2 p . The relevant field equations can be obtained by varying the action with respect to metric and connection independently, and then one obtains after a trivial but rather lengthy calculation the field equations which are given in Appendix A.
In the following section we are going to fix the free parameters which appear in the action due to its general form of definition.

Conformal invariance of the action
The action in the form of (9) has free parameters that must be fixed. This can be done by imposing some constraints or initial conditions where, we use the conformal invariance condition. The first extension of Einstein's gravity was done by Weyl in 1919 and then developed by Cartan and Dirac (for review see [2,22]). In [23], conformal invariance was considered in Riemann-Cartan geometry where the torsion plays the role of an effective Weyl gauge field. Conformal torsion gravity and conformal symmetry in teleparallelism were studied in [24].
Under the conformal transformation of metric of the form the non-metricity which is associated with ∇, transforms as where Ω is a scalar function of x. However, the conformal transformation of the torsion tensor is somehow ambiguous and different forms of transformation are considered; namely, Weak conformal transformation under which the torsion tensor remains unchanged and Strong conformal transformation in which the torsion tensor transforms similar to (10), namely one has: S µν λ = Ω 2 S µν λ [10]. Being interested in conformal invariance of the action, we choose the weak form which preserves the anti-symmetric part of the affine connection. In accordance with the all above mentioned, one can easily show that the affine connection is not changed under the conformal transformation of the form (10).
It is straightforward to show that under conformal transformation, the action (9) transforms in the following way: ln Ω + 4a 7 + 4na 8 + 4a 9 + 4na 10 + 4n 2 a 11 ∇ λ ln Ω (∇ λ ln Ω) (12) in which ∇ 2 ln Ω = ∇ {λ} ∇ {λ} ln Ω. The conformal invariance of the action results in vanishing the additional terms at the above relation, that leads one to write: Inserting (13) in (9) leads to This is the general form of the Weyl-invariant metric-affine gravity (WMAG), noting that a compensating Weyl scalar is needed to cancel out the Ω n−2 factor which appears in (12).
More simplification can be done if one is interested in the reduced form of metric-affine space, for example, in Einstein-Weyl-Cartan space one has: ∇ λ g µν = −2A λ g µν , where A λ is a vector filed [27]. By applying this condition to (14) it turns to the following simple form We would like to mention that (15) is expressible in a special form when the affine curvature scalar and covariant derivative are redefined in terms of the Riemannian part plus a particular combination of torsion and non-metricity square-terms by using (6), (7) and (8). Therefore, the affine curvature scalar takes the form of and in Einstein-Weyl-Cartan space it becomes With the same procedure one obtains and substituting (17) and (18) in (15) results in Now by imposing the torsion-free condition on (19) and setting a 0 = 1, up to a compensating Weyl scalar it becomes similar to what studied as a conformally invariant extension of Einstein-Hilbert action [28].
It is worth noting that when the action in (14) is transferred to Weitzenböck space by setting the curvature and non-metricity to zero [7], it reduces to which is similar to the one used in teleparallel theories of gravity [19,29]. For example, in Ref. [30], from the tetrad and a scalar field analysis of torsion, it is shown that such an action with specific coefficients of a 1 = 0, a 2 = − 1 3 , a 3 = 1 2 and a 4 = 1 4 , can indeed be a conformally invariant teleparallel action.

Conclusion
In this paper we have first studied the general form of second order metric-affine action which is constructed from all the possible forms of 15 scalar terms made up of affine curvature, torsion and nonmetricity tensors. The Weyl-invariant extension is obtained by imposing conformal invariance as a condition on the action. The resultant action reduces to the conformally invariant teleparallel action in transition to Weitzenböck space [30]. Studying conformally invariant torsion theories are important because aside from the conformal invariance property, they can address some important issues of theoretical physics (for example see [31]). Torsion theories may be viewed as a rank-3 mixed symmetry tensor field. From the space-time symmetry and group theoretical point of view, it is proved that linear conformal quantum gravity in flat and de Sitter backgrounds should contain such mixed symmetry tensor filed of rank-3 [32,33,34]. Our obtained action (9) and its Weyl-invariant extension covers the theories that made up from Riemann curvature equipped with the torsion and non-metricity, albeit studying the Weyl invariance of such theories needs some modifications, however as discussed, in our method the Weyl invariance can only be obtained by fixing the coefficients, that we have considered it as a special case.

Acknowledgement
The author would like to thank M.V. Takook for his helpful guidelines.
Variation of the action with respect to the connection yields the following equation: