A New Class of Ghost and Tachyon Free Metric Affine Gravities

We construct the spin-projection operators for a theory containing a symmetric two-index tensor and a general three-index tensor. We then use them to analyse, at linearized level, the most general action for a metric-affine theory of gravity with terms up to second order in curvature, which depends on 28 parameters. In the metric case we recover known results. In the torsion-free case, we are able to determine the most general six-parameter class of theories that are projective invariant, contain only one massless spin 2 and no spin 3, and are free of ghosts and tachyons.


Introduction
Metric-Affine Gravity (henceforth MAG) is a broad class of theories of gravity based on independent metric (or tetrad) and connection. The study of MAG has a long history [1,2]. A general linear connection will have torsion and non-metricity. In the literature, more attention has been given to theories with torsion, but recently there has been a great deal of interest for MAGs with non-metricity, see e.g. [3][4][5][6][7][8][9][10][11][12].
There can be many reasons to study such theories. The main reason for our interest in MAG is its relation to quadratic gravity 1 and its similarity to gauge theories of the fundamental interactions. Quadratic gravity is known to be renormalizable [13] and asymptotically free [14] but prima facie not unitary, as expected of a theory with a kinetic term with four derivatives. There have been many proposals to circumvent this problem, but none has proven entirely convincing [15][16][17][18]. More recent progress has been reported in [19][20][21]. In spite of this, there has been a revival of interest in quadratic gravity, especially in connection with the possibility of realizing scale invariance at high energy [22][23][24][25][26].
MAG is closely related to quadratic gravity, since it can be rewritten as quadratic gravity coupled to a specific matter type. Let A denote a general linear connection and F its curvature; also, let Γ be the Levi-Civita connection and R its curvature. Splitting A = Γ + φ, where φ is a general three-index tensor, an action of the form (F + F 2 ) becomes, schematically R + φ 2 + (R + ∇φ + φ 2 ) 2 . (1.1) In this way one can study large classes of theories of gravity and matter with special geometrical features. 2 In MAG the kinetic terms contain only two derivatives, but ghosts are still generically present, due to the indefiniteness of the quadratic form F 2 . Thus, much of the discussion that is going on for quadratic gravity could be applied also to MAG. However, the status of MAG is much less understood.
It is thus of obvious interest to determine what special classes of MAGs could be free of ghosts and tachyons. In the metric case, the most general ghost and tachyon-free theories not containing accidental symmetries 3 have been determined in [28,29]. It was based on the use of spin projectors for a general two-index tensor and a three-index tensor, antisymmetric in one pair. 4 A more detailed analysis of a large number of cases including also accidental symmetries has been given recently in [30]. A broader analysis of the spectrum of a Poincaré gauge theory has been given in [31], where a class of ghost-and tachyon-free models were obtained. The purpose of this paper is to give the tools that are necessary to address this problem for general MAG, containing both torsion and non-metricity, and to exhibit a new class of ghost-and tachyon-free theories with non-metricity.
The relation of MAG to gauge theories of fundamental interactions is best understood if one uses arbitrary frames in the tangent bundle. The theory is then seen to have a local gauge invariance under diffeomorphisms and under local GL(4) transformations, but it is in a Higgs phase [32][33][34][35]. The frame field, the metric and the connection are all independent, with the first two playing the role of Goldstone bosons. The gauge GL(4) is "spontaneously broken" to the trivial group and the connection (or more precisely the difference between the connection and the Levi-Civita connection) becomes massive.
This formalism is not well-suited for practical applications because it contains a large number of redundant fields (essentially, the 16 components of the frame field). In a linearized analysis one would discover that these fields are all part of the kernel of the kinetic operator and can be gauge-fixed to be zero. It is convenient instead to work from the start with a formalism that contains the smallest number of fields. This is the standard formulation in terms of a metric g µν and an independent connection A λ µ ν . In this formalism the only gauge freedom is the diffeomorphism group and one cannot reduce the number of fields further while preserving locality. 5 It is important, however, to keep in mind that this is just a gauge-fixed version of the general GL(4) formulation, and is gauge equivalent to the vierbein formulation.
In the following we start from the most general MAG action which contains 28 free parameters, and determine the conditions under which it has additional symmetries under shifts of the connection. We then determine the spin projection operators for the fields that appear in the linearized action, which facilitate the inversion of the wave operator to obtain the propagator for each spin sector. We then specialize these results to the case of theories with metric or torsion-free connections. In the latter case we determine a six-parameter family of theories that are ghost-and tachyon-free, propagating a massless graviton and massive spin 2 − , 1 + and 1 − states with distinct masses.

.1 The action
In the model we shall consider, the independent dynamical variables are the metric g µν of signature − + . . . + and a linear connection A µ ρ σ The curvature is defined as
whereas torsion and non-metricity are defined by 6 As an action we take Note that there are two "pseudo-Ricci" tensors F (13) µν and F (14) µν , without symmetry properties, and one "pseudo-Ricci scalar" that we denote F . The Einstein-Hilbert action is described by the a 0 g µν F (13) µν term. The action contains 28 parameters, namely (a 0 , a 1 ,...,a 11 , c 1 ,...,c 16 ). In d = 4, however, the combination which reduces to the Gauss-Bonnet integrand in the Riemannian case, does not contribute at quadratic level when expanding around flat space. Indeed, in Weyl geometry (i.e. if the nonmetricity is of the form Q λµν = v λ g µν ), it is a total derivative [37]. In the presence of tracefree non-metricity, it is not a total derivative [38], but in flat space it only gives cubic and quartic interactions . Thus, for the purposes of our analysis, one parameter is redundant. Turning to the action (2.4), it is convenient to express it as The tensors G and H inherit the symmetries of the objects they are contracted with. Furthermore, G, A and B are also symmetric under the interchange of the first half of indices with the second half. In the following expressions, symmetrizations that are not already manifest are indicated: 7 , (2.9) where it is understood that G is to be symmetrized with respect to interchange of indices (µ 1 ...µ 4 ) and (ν 1 ...ν 4 ), and that A, B and C with respect to the interchange of indices (µ 1 ...µ 3 ) and (ν 1 ...ν 3 ).

Gauge symmetries
In general the action is invariant under the action of diffeomorphisms, For an infinitesimal transformation x ′µ = x µ − ξ µ (x) the transformation is given by the Lie derivatives, plus an inhomogeneous term for the connection: In four dimensions, if all the coefficients a i are zero, the action is additionally invariant under the following realization of Weyl transformations: This is the usual way in which Weyl transformations are realized on Yang-Mills fields, while the Levi-Civita connection transforms as: In the following we shall be interested in cases where the action is invariant under additional transformations of the connection (see also [39]). The following three classes of transformations will be relevant. First we consider the projective transformations where λ µ (x) is an arbitrary gauge parameter. Under this transformation In particular δ 1 F = 0. Assuming that neither torsion nor the non-metricity vanish, one finds that the action is invariant provided that There is a similar transformation with the second index singled out under which In this case, the variation of the general action gives rise to a large number of independent structures. Then the invariance of the action requires that Finally there is the transformation that singles out the third index under which Once again, the variation of the general action gives rise to a large number of independent structures. Assuming that the torsion and non-metricity do not vanish, the action is invariant provided that 3 Linearization and spin projectors

Linearized action
The equations of motion that come from the action (2.4) have as a solution the Minkowski space Expanding the action around this solution, the quadratic wave operator takes the form where, by abuse of notation, we denote A also the fluctuation, and This operator has a kernel consisting (at least) of the infinitesimal diffeomorphisms (2.13), which in the present case read For specific values of the couplings the kernel could be larger.

Spin projectors
In the analysis of the spectrum of operators acting on multi-index fields in flat space, it is very convenient to use spin-projection operators, which can be used to decompose the fields in their irreducible components under the three-dimensional rotation group [40][41][42]. For a three-index tensor that is antisymmetric in one pair of indices, the spin projectors were given in [28,43]. The spin projectors for totally symmetric three-tensors have been given also in [44]. To the best of our knowledge, the spin projectors for a general three-index tensor have not been given in the literature. We thus turn to the construction of these objects.

GL(d)-decomposition
The space of two-index tensors can be decomposed into irreps of the group GL(d), given by symmetric and antisymmetric tensors. The projectors onto these subspaces are (3.5) The finer decomposition into irreps of SO(d − 1) is widely used in gravity. The corresponding treatment of three-index tensors is algebraically more complicated. We begin with some elementary facts about three-index tensors as representations of GL(d). In order to discuss their symmetry properties, we will focus on the second pair of indices. Thus when we say that t cab is (anti)symmetric, without further specification, we mean t cba = ∓t cab .
The space V of three-index tensors has dimension d 3 . The subspaces V (s) and V (a) of symmetric and antisymmetric tensors are invariant subspaces of dimensions d 2 (d + 1)/2 and d 2 (d − 1)/2 respectively. The projectors onto these subspaces are (3.6) The subspaces V (ts) and V (ta) of totally symmetric and totally antisymmetric tensors are invariant subspaces of dimension d(d − 1)(d − 2)/6 and d(d + 1)(d + 2)/6 respectively. Given any tensor, one can extract its totally (anti) symmetric part by means of the projectors The complement of V (ts) in V (s) and of V (ta) in V (a) are also invariant subspaces denoted V (hs) and V (ha) respectively. 8 They consist of tensors that are (anti) symmetric but have zero totally (anti) symmetric part. The projectors onto such subspaces are Thus the decomposition of a three-index tensor in its GL(d)-irreducible parts is . In physical applications q a has the meaning of a four-momentum. Given q a , we can decompose every other vector in parts longitudinal and transverse to it, by using the projectorŝ This leads to a finer decomposition of V into irreps of the group SO(d − 1). As a first step we expand the identity in eight terms. It is easy to see that the combinations (all with fixed indices) are projectors. Then consider the simultaneous eigenspaces with eigenvalue 1 of these and of the GL(d) projectors introduced above. The dimensions of these spaces are given in Table 1. The last column and the last row give the total dimension of the +1 eigenspaces of the projectors in the corresponding rows and columns.
All of these spaces are representations of SO(d−1), some irreducible and others not. In order to obtain the irreps, let us note that the hs-and ha-projections of 3 2 LT T and T T L+T LT − 1 2 LT T are themselves projectors. Finally, in several of these representations one can isolate the "trace" and the "tracefree" part. In dimension d = 4, the SO(3) irreducible representations are then ts hs ha ta dim Table 1: Dimensions of projected spaces in d dimensions.  Table 2, together with the spin and parity carried by them. For completeness we also list the representations carried by the two-index symmetric tensor h. The subscripts refer to the number in the labelling of the projectors.
A given representation of the group SO(3) may appear more than once in the decomposition of A cab . These copies will be distinguished by a label i. Thus for example the representation 2 − occurs twice, and the two instances are denoted 2 − 1 and 2 − 2 . In addition, the same representation may occur also in the decomposition of the two-tensor h ab . We use the same label for all these representations. Thus for example the representation 2 + occurs altogether four times: the representations 2 + i with i = 1, 2, 3 come from A cab whereas 2 + 4 comes from h ab . For each representation J P i there is a projector denoted P ii (J P ). In addition, for each pair of representations with the same spin-parity, labelled by i, j, there is an intertwining operator P ij (J P ). We collectively refer to all the projectors and intertwiners as the "spin-projectors". Formulas for all the spin projectors are given in Appendix A. For convenience they are also given in an ancillary Mathematica notebook on the arXiv.
Let us emphasize again that these spin projectors are suitable to decompose tensors that either have no symmetry property or are (anti)symmetric in the last two indices. If one is interested in tensors that are (anti)symmetric in the first and third index, it is more convenient to work with another set of spin projectors P ′ ij (J P ), such that whenever the representation i or Table 3: Count of fields of general MAG: irreps of given spin contained in A (2nd column) in h (3rd column); their total number (4th column) and total number of fields they carry in d = 4.
j is carried by a three-index tensor, the first two indices are permuted. For example Similarly one can deal with tensors that are (anti)symmetric in the first two indices.

Rewriting the quadratic action
The projector P ij (J P ) has two sets of hidden indices: one for the representation J P i and one for the representation J P j . These multi-indices A, B... consist of either three or two indices, depending whether the carrier field of the representation is A or h. Thus for example P 11 (2 + ) has indices P 11 (2 + ) cab def , P 41 (2 + ) has indices P 41 (2 + ) ab def etc. The spin projectors satisfy the orthonormality relation 15) and the completeness relation The linearized quadratic action (3.3), can be written as In four dimensions, the kinetic operator is an 74 × 74 matrix, that we have written as 64 × 64, 10 × 10 and off-diagonal 10 × 64 and 64 × 10 blocks. Since the operator is Lorentz-covariant, it maps states of a given spin and parity to states of the same spin and parity. Therefore, decomposing A cab and h ab into irreducible representations of the rotation group puts the kinetic operator in block diagonal form.
Expanding the operator O AB in terms of these projection operators one can rewrite the quadratic action as Exploiting the relations (3.15,3.16), the matrix elements a ij (J P ), where both representations J P i and J P j are carried by A, can be obtained by for any fixed k, and where d(J P ) is the dimension of the representation J P . The second equality follows from (3.15), and it shows that it suffices to know the projections operators P jk for any fixed k in order to obtain all coefficients matrices. This was also observed in [31], where P jk for a fixed k ( chosen for convenience to give the simplest projector) were referred to as 'semiprojectors'. Similarly, if the representation J P i is carried by A and J P j is carried by h, we can use for example where we have chosen k that is carried by A. These matrices a ij (J P ) will be referred to as the "coefficient matrices". For a general MAG in four dimensions, they are given in Appendix B.1.

Constraints for ghost-and tachyon-freedom
Let us arrange the fluctuations into a multi-field Φ A and introduce corresponding sources: Adding source terms, the linearized action can be written which gives the field equations JP ij Inverting for Φ as a function of J and substituting back into S (2) we obtain a quadratic form in J that we identify with the saturated propagator and we denote by Π. There is, however, a complication: in a given spin-parity sector, the matrix a ij may have null eigenvectors. This corresponds to the presence of gauge symmetries as follows. Suppose for a given J P , the matrix a ij is n × n and has rank m, thereby admitting (n − m) null vectors, Then (4.2) is easily seen to be invariant under where ξ (r) are arbitrary functions of the coordinates, provided that the sources obey the constraints The preceding analysis has to be repeated in each spin sector to determine all the gauge symmetries and source constraints. In practice this cumbersome procedure will not be necessary for the following reasons.
Let us distinguish gauge symmetries that are already present in the original action (2.4) from "accidental" symmetries that are only present in the linearized action. The latter are broken by interactions and therefore cannot be maintained in the quantum theory. In the following we shall restrict ourselves to theories that do not have accidental symmetries. Thus, the only infinitesimal gauge invariance is given by the diffeomorphisms (3.4): Writing this schematically as δΦ = Dξ, since Dξ is a null eigenvector of the linearized kinetic term, we must have Explicit calculation shows that P ij (J P )Dξ is only nonzero for J P = 1 − and j = 4, 5, 6, 7 or J P = 0 + and j = 4, 6. Then one finds that a(1 − ) has the null eigenvector and a(0 + ) has the null eigenvector (0, 0, 0, i|q|/2, 0, 1). Thus, in general, the ranks of the coefficient matrices a(1 − ) and a(0 + ) are 6 and 5, respectively. Invariance of the source term then demands that the sources satisfy the constraint 9 2iq a σ ac + q a q b τ bca = 0 . To obtain the propagator sandwiched between physical sources one takes the inverse of any m × m submatrix of a ij with nonzero determinant. This amounts to fixing the gauge symmetries and it does not effect the form of the physical saturated propagator [45]. Denoting this submatrix by b kℓ , (k, ℓ = 1, ...m), the resulting saturated propagator Π, upon solving for Φ in terms of the source and substituting back into the action, takes the form where C kℓ is the transpose of the cofactor matrix associated with the matrix b, which is assumed to have rank m. It is important to stress that in our notation b −1 kℓ denotes the matrix element of b −1 in the representations k, ℓ, which need not agree with the element of the matrix b −1 in the k-th row and ℓ-th column (unless a is non-degenerate, in which case b = a). Given that b ij (q) is a hermitian matrix and its momentum dependence is polynomial, the poles at non-vanishing values of q 2 can only come from det b(J P ). We assume that for each given J P there will be s propagating particles, with s ≤ m. Then we can write where (C, m 2 1 , . . . , m 2 s ) are constants. For a physical spectrum these constants must be real and to simplify the analysis we shall further assume that the masses m 2 n , n = 1, ..., s, are nonvanishing and distinct (possibly, one of the masses could be zero). The determinant det b has a simple zero for q 2 = −m 2 n , so exactly one eigenvalue of b must have a zero there. This implies that the residue matrix lim has exactly one non-vanishing eigenvalue.
Before proceeding to implication of this for ghost-freedom criteria, we need to first note that the spin projectors in (4.11) contain powers of 1/q 2 that do not contribute to the physical propagators. These spurious poles at zero momentum, which we shall sometimes refer to as kinematical singularities, cancel out in the full saturated propagator. These poles arise from the product of constants, or 1/(q 2 + m 2 ), with the longitudinal parts of the spin projection operators. In the latter case, the simple procedure of partial fractions gives rise to terms in which the spin projection operator are evaluated on the mass shell, plus terms with powers of 1/q 2 . For example: and similarly for expressions of the form 1/((q 2 ) n (q 2 + m 2 )). The first term on the r.h.s. has the same pole at q 2 = −m 2 , but in its coefficient the momentum squared is now evaluated at the pole. The second term gives another spurious pole at zero. In the end all the spurious poles cancel out and we remain with a combination of the spin projectors evaluated on the mass shell or constants sandwiched between sources that obey source constraints.
With the issue of kinematical singularities out of the way, we can now state the conditions for the absence of ghosts and tachyons. The tachyon-freedom condition is very simple, namely Tachyon-free =⇒ m 2 n > 0 , n = 1, ..., s .

(4.15)
To examine the ghost-freedom condition, it is convenient to diagonalize the matrix b −1 . Denoting its eigenvalues by λ I , and the corresponding eigenvectors by V (I) , we have where J Ghost-freedom requires that for each value of k the residue of the sum in (4.16) must be negative. As already remarked, precisely one eigenvalue has nonzero residue at a given pole. Thus, noting also that the modulus of the source-squared term evaluated at q 2 = −m 2 n is finite, we can express the ghost freedom condition as 10 Going back to the formula (4.11), or (4.16), in any J P sector involving the matrix b −1 with rank greater than one, there will clearly be mixing of sources that survive the source constraints. Given that all the kinematical singularities have cancelled, the result for the saturated propagator in such J P sectors can be written in such a way that the standard form of the spin J P propagators arise in terms of a suitable combination of these sources. This phenomenon will be clearly shown in the multi-parameter models analysed below; see (5.9) and (6.34).
Given any MAG with specific couplings c 1 . . . c 16 , a 0 , a 1 . . . a 11 one can use these conditions on the coefficient matrices given in Appendix B.1, and determine the spectrum of the theory. However, the 28-parameter class of all MAGs is too broad for a general analysis, so in the following we discuss two important subclasses: MAGs with either Q = 0 or T = 0.

General case
In metric theories the following identities hold: Using these properties, the most general action up to and including curvature and torsion squared terms is a 10-parameter action given by Note that the metricity condition Q = 0 is a kinematic constraint that changes the nature of the theory: the action (5.2) is not obtained from the general MAG action (2.4) simply by specializing the values of the couplings. Nevertheless, it is useful to write it in the same form and to preserve the numbering of the invariants. To distinguish the two cases, we changed the name of the couplings from c i to g i and from a i to b i . Notwithstanding the fact that the action (5.2) is not a special case of (2.4), it is possible to linearize it by making use of the results already computed for the general action (2.4) as follows. Let us first consider the F 2 terms. In the action (5.2), and in accordance with (5.1), making the substitutions and comparing the result with the general action (2.4), we obtain the relations Next, let us consider the substitution required for the parameters a i in terms of b i . This is more subtle due to the fact that expanding around A cab = 0, the variation of the metricity condition implies that the fluctuation fields are related by where we recall that A denotes also the fluctuation. Thus inserting in the linearized action the decomposition A cab = A c[ab] + A c(ab) , the symmetric part of A gives terms proportional to h that can be compared to those that, in a general MAG, are produced by Q. This gives the relations In summary, the coefficient matrices of the metric theory are obtained from those of the general MAG by inserting the values for the couplings c i , a i in terms of g i , b i as given in (5.4) and (5.6), and deleting all the rows and columns that pertain to representations carried by symmetric three-tensors. The remaining representations, and the count of degrees of freedom that they carry, is given in Table 4. The coefficient matrices of metric MAG in d = 4 are given explicitly in Appendix B.2.

Neville's model
In order to test of our formulae and procedures we reconsider here, as an example, the Neville model [43], which is the same as model (ii) in [28]. It corresponds to choosing the couplings g 1 = g 3 = −g 4 /4 ≡ g, g 7 = g 8 = g 16 = 0, b 1 = b 2 = b 3 = 0.
In the sectors 1 − and 0 + , to fix diffeomorphism invariance, we choose the non-degenerate b-matrices to be the upper left 2 × 2 sub-matrices of the general a-matrices given in Appendix B.2, namely b −1 ij (1 − ) with i, j = 3, 6 and b −1 ij (0 + ) with i, j = 3, 5. The inverses of these coefficient matrices are then given by: The analysis of section 4.4 shows that this theory contains a massless graviton and a massless pseudoscalar state, with mass m 2 = a 0 /(6g). Absence of tachyons and ghosts requires a 0 > 0 and g > 0. The saturated propagator is: As discussed in section 4.4, and using the source constraint (4.10), it can be rewritten in a more explicit form, where the spin projection operators are put on shell: where S ab = σ ab + iq c τ acb σ ab , and following [28] we have used The last term is the standard graviton propagator while for the spin 0 − we have The spin 1 + and 1 − contributions actually vanish.

.1 General case
In torsion-free theories the following identities hold: [µν] . (6.1) These reduce the number of independent invariants. One finds that the terms in (2.4) with parameters c 5 , c 6 , c 13 , c 14 , c 15 , a 1 , a 2 , a 3 , a 9 , a 10 , a 11 become redundant. Thus we parameterize the most general torsion-free MAG action as  Once again we note that T = 0 is a kinematic constraint, so that the theories we now consider are not equivalent to just setting to zero the parameters listed above. For this reason, the remaining parameters c i have been renamed h i .
In the torsion-free case, the field A λµν is symmetric in λ, ν. In four dimensions, this reduces the number of degrees of freedom of A from 64 to 40. The corresponding spin representations are listed in the second column of Table 5. In order to obtain the coefficient matrices, we use the "primed" spin projectors defined in the end of section 3.2, which are better suited to decompose a tensor symmetric in the first and last index. All the primed spin projectors in the columns ha and ta in Table 2 give zero when acting on a torsion-free connection. Thus, the coefficient matrices for this case are smaller: their dimensions are given by the fourth column of Table 4. A diffeomorphism (2.12) preserves the symmetry of A λµν and diffeomorphism symmetry reduces by one the rank of the coefficient matrices for spins 1 − and 0 + . The coefficient matrices for the torsion-free theory in four dimensions are given in Appendix B.3.

Torsion-free theories with projective symmetry
Let us now examine the possible additional symmetries in this case. We find that while the symmetry (2.19) is still too restrictive, in the sense that it requires all c-coefficients to vanish, we can achieve projective symmetry, which is now a symmetric combination of (2.16) and (2.22): It follows that Invariance of the action is found to require that where we have used (6.1) and the following formula with a total derivative term discarded. The part of the action proportional to h 2 , vanishes due to the identity which follows from repeated use of the second equation in (6.1). Therefore, the action depends on nine parameters, namely, (a 0 , a 6 , a 7 , a 8 ) and (h 7 , h 8 , h 11 , h 12 , h 16 ), and it takes the form where the parameters (γ 1 , ..., γ 7 ) are defined in terms of the 9 parameters of the action as In four dimensions, the projective symmetry eliminates four fields, reducing by one the ranks of the coefficient matrices 1 − and 0 + . In fact one finds that a(1 − ) has the null eigenvectors 10/3 i|q|, 2/3 i|q|, 3/2 i|q|, 0, 1 , √ 10, √ 2, √ 2, 1, 0 , (6.10) while a(0 + ) has the null eigenvectors The ranks of the coefficient matrices for the representations 3 − , 2 + , 2 − , 1 + , 1 − , 0 + are 1, 3, 1, 1, 3, 3 respectively.
Invariance of the source term implies that the sources must obey the constraints: Next, we examine the spectrum of this 9-parameter model.

New ghost-and tachyon-free theories
To further simplify matters, we shall restrict our attention to choices of parameters such that: (i) Spin 3 field does not propagate, and (ii) in the spin 2 + sector only the massless graviton propagates.
Demanding that the determinant of this matrix contains no powers of −q 2 higher than one, leads to h 12 = −h 11 , h 8 = −h 7 . (6.14) With these conditions, the class of actions that we consider are of the form [µν] F (14)µν where we introduced the following convenient new combinations of parameters: Let us now discuss the dynamical content of this theory. We have already ruled out the propagation of a spin-3 state, for which In the spin-2 + sector we have As is well known, the propagation of a massless spin 2 + state requires an admixture of a spin 0 + state. Having imposed (6.13) and (6.14), and fixing the diffeomorphism-and projective-gauge by choosing the non-degenerate coefficient submatrix to be b ij (0 + ) with i, j = 3, 4, 5, we get Thus, the existence of a massless graviton requires that A, B, C and a 0 are all nonvanishing. In particular, this implies that the coefficient matrix for the spin 3 sector is not zero.
As we shall now see, having imposed (6.13) and (6.14) we find that all the coefficient matrices have maximum rank submatrices whose determinants are at most first order in q 2 . This means that in any given sector at most one state propagates. Indeed, denoting b(2 − ) = a(2 − ) 11 , b(1 + ) = a(1 + ) 11 , and taking the non-degenerate submatrix b ij (1 − ) with i, j = 2, 4, 5, we find b(2 − ) = 1 4 2B + (30h 7 + 9h 11 ) q 2 . (6.20) b(1 + ) = 1 6 3B + (40h 7 + 17h 11 ) q 2 , (6.21) Note that since A, B, C are nonvanishing, there is no room for accidental symmetries. From these equations we read off the masses of the modes 2 − , 1 + and 1 − : , (6.23) We can now list the matrices b −1 ij (J P ): We can now state the ghost-and tachyon-free conditions. The tachyon-free conditions amount to the positivity of the masses (6.23), which are equivalent to (10h 7 + 3h 11 )B > 0 , (40h 7 + 17h 11 )B > 0 , BC(16B + 25C)(2h 7 + h 11 ) > 0 . (6.30) Applying the formula (4.18), one finds that the ghost free conditions for the spin 2 + , 2 − , 1 + and 1 − sectors, are given by a 0 > 0 , 10h 7 + 3h 11 < 0 , 40h 7 + 17h 11 < 0 , (2h 7 + h 11 ) < 0 . (6.31) All these conditions together are equivalent to a 0 > 0 , B < 0 , C(16B + 25C) > 0 , and Finally, the saturated propagator is We can make this expression more understandable by explicitly displaying the denominators of each propagator, and evaluating the contractions of the spin projectors with the sources: where we defined and div 1 τ ab = q c τ cab , div 2 τ ab = q c τ acb , div 12 τ a = q b q c τ cba , div 13 τ a = q b q c τ cac , tr 13 τ a = τ ca c . (6.37) This manifestly shows the spin-2 + , 1 + and 1 − degrees of freedom being sourced by suitable combinations of sources. In particular we note that the spin 2 + and 1 − degrees of freedom have propagators of the standard form. The propagator for the spin 1 + seems less familiar, but it is simply that of a massive two-form potential, described by the Lagrangian where H µνρ = 3∂ [µ B νρ] . We also note that, unlike the case of spin 2 + , the spin 2 − propagator cannot be written solely in terms of second rank tensor sources, as it necessarily requires the presence of the 3rd rank sources.

Conclusions
In this paper we have set up the machinery that is necessary to analyze the spectrum of a general MAG theory. In particular, we have constructed the spin projectors for a general three-index tensor, and we have used them to rewrite the wave operator for the most general, 28-parameter MAG. Not surprisingly, this case turns out to be too complicated to determine its spectrum, but it is possible to do so in special subclasses of theories. We have considered here theories that have either vanishing non-metricity, recovering previously known results, and theories with vanishing torsion. In the latter case the theory depends on 17 parameters; imposing projective invariance reduces this to 10 parameters and imposing that there be no propagating spin 3 − and no massive propagating spin 2 + fields, further reduces this number to 6. Absence of ghosts and tachyons results in the inequalities (6.32) on these six parameters. Even within the torsion-free subtheory, relaxing the conditions of section 6.3 will lead to a much more complicated system.
With hindsight, the absence of ghosts and tachyons in these models is related the fact that, when converting to the R, φ variables in the manner of equation (1.1), they do not contain any terms quadratic in curvature. For the same reason, these models are also non-renormalizable. This is entirely analogous to the situation also pointed out in [28] for the nine parameter metric quadratic theories with torsion. Similarly, we expect that allowing a propagating massive spin 2 + mode will probably make the theory renormalizable but not unitary.
It is important to stress that the metric and torsion-free cases are kinematically distinct from the original general MAG and that the ghost-and tachyon-free models we have found are not special cases of the general MAG, but only of the kinematically restricted models. In fact, some classes of ghost-and tachyon-free Poincaré gauge theories have been found in [31] that are different from our six parameter ghost-and tachyon-free model. We leave it for future work to study special subclasses of the general MAG.
Also of some interest would be the study of models with propagating spin 3 − . It is known that the free massless spin 3 theory can be embedded in linearized MAG [46], however the underlying linearized gauge symmetry does not extend to the full theory. It would be interesting to explore whether MAG can describe a massive spin 3 field coupled to gravity. We hope to return to these questions in the future.
Note added: After this work appeared on the arXiv, we have been informed that the spin projectors for the general theory have also been worked out in [48], and that they agree with ours.

A Spin projectors
In the torsion-free case the spin projectors have also been given in [5].
The negative parity projectors are given by where it is understood that there is no summation over the indices displayed, and Note that the transposition raises and lowers the vector indices on T and L such that, for example, T c d and T ca get mapped to T d c and T ca , respectively. Therefore, we have (B T ) def cab = A.2 P (J + ) projectors, J = 0, 1, 2 The positive parity projectors are given by B The coefficient matrices

B.1 General MAG
Here we provide the coefficient matrices a(J P ) arising in the expansion of the wave operator in the general 28 parameter model, in terms of the spin projection operators. As a weak check, we observe that all coefficient matrices vanish identically for the combination (2.6).

B.2 Metric MAG
Using (5.4) and (5.6) in the coefficient matrices of Appendix B.1, and deleting the rows and columns that pertain to symmetric three-tensors, we recover the coefficient matrices of the metric theory, as computed in [28] (see also [47]) with two differences. First, one has to keep in mind that the graviton field h ab used here is equal to one half of the graviton field ϕ ab used in those references. This gives a factor 2 in the mixed A-h coefficients and 4 in the h-h coefficients. Second, the projectors P ij (1 + ) with i, j = 2, 3 span the same space as P ij (1 + ) with i, j = 1, 2 in those references, but differ by a linear transformation (The old projectors do not respect the GL(4) decomposition). This is of no consequence for the physical results.

B.3 Torsion-free MAG
We give here the coefficient matrices for the most general torsion-free model, as discussed in section 6.1. If one wishes to further impose the projective symmetry discussed in Sect. 6.2, one has to further impose (in four dimensions) the conditions given in equation (6.5).