Higher-spin gravity and torsion on quantized space-time in matrix models

A geometric formalism is developed which allows to describe the non-linear regime of higher-spin gravity emerging on a cosmological quantum space-time in the IKKT matrix model. The vacuum solutions are Ricci-flat up to an effective vacuum energy-momentum tensor quadratic in the torsion, which arises from a Weitzenb\"ock-type higher spin connection. Torsion is expected to be significant only at cosmic scales and around very massive objects, and could behave like dark matter. A non-linear equation for the torsion tensor is found, which encodes the Yang-Mills equations of the matrix model. The metric and torsion transform covariantly under a higher-spin generalization of volume-preserving diffeomorphisms, which arises from the gauge invariance of the matrix model.


Introduction
Our present understanding of gravity in terms of general relativity (GR) is incomplete for several reasons. One problem is that GR is not renormalizable [1], and hence does not define a unique quantum theory. This is reflected in notorious difficulties trying to quantize various formulations of GR. Broader approaches towards a quantum theory of gravity include notably string theory, which leads to gravity in 10 dimensions. However, ad-hoc reductions to 3+1 dimensions lead to a lack of predictivity known as the landscape problem. Since gravity is tied to the structure of space-time, a natural strategy is to develop a suitable "quantum" framework for space-time, based on generalized or noncommutative notions of geometry. However, there is little reason to expect that straightforward attempts to mimic GR in such a framework would overcome these issues. Moreover, pathological long-distance effects arise generically on non-commutative spaces due to virtual string-like non-local modes, known as UV/IR mixing [2][3][4]. They cancel only in the maximally supersymmetric IKKT matrix model [5], which should thus have the best chance to describe physics, leading to an unexpected link with string theory. However, it is not obvious how to obtain gravity from Yang-Mills-type matrix models such as the IKKT model. There are intriguing hints such as non-local gauge transformations and Ricci-flat propagating metric fluctuations [6][7][8], but the presence of an anti-symmetric tensor θ µν in space-time leads to a dangerous breaking of Lorentz invariance. This problem can be overcome by considering a higher-spin generalization, where θ µν is replaced by a twisted bundle of such tensors over space-time. This is realized in a simple solution of the IKKT model with a mass term interpreted as cosmological FLRW space-time M, based on the doubleton representations of so(4, 2) [9,10], cf. [11]. It leads to a higher spin gauge theory which is invariant under a higher spin generalization of volume-preserving diffeomorphisms, ghost-free at the linearized level, and includes spin 2 gravitons. A linearized Schwarzschild-like solution was also found [12]. The theory has similarities with Vasilievs higher-spin gravity [13,14] but also clear differences; notably it is defined through an action, and there are 5 propagating graviton modes. In the present paper, we study that higher-spin theory at the non-linear level. This is not straightforward because the model is of Yang-Mills type, there is no Einstein-Hilbert action, and everything is based on Poisson brackets (or commutators). The gauge symmetry corresponds to generalized diffeomorphisms rather than local Lorentz transformations 2 , hence it is not some reformulation of GR in the spirit of MacDowell-Mansouri [16]. A more appropriate approach can be found in a paper by Langmann and Szabo (LS) [17], who pointed out that a dimensional reduction of a gauge theory in 8-dimensional phase space with suitable constraints can be interpreted in terms of 4-dimensional teleparallel gravity through torsion. Although their setup does not provide a complete theory, a similar strategy provides a geometric understanding of the present model. See also [18,19] for related work. The first message of the present paper is indeed that torsion, rather than curvature, is the key to a geometric understanding of the matrix model. The reason is quite simple: the matrices Z a of the model naturally define a frame E a = [Z a , .], which defines the effective metric governing all propagating modes in the model. Given this intrinsic frame, it is natural to consider the associated Weitzenböck connection ∇, which is defined by ∇E a = 0. This connection has no curvature but torsion. The crucial observation is that this torsion naturally encodes the field strength [Z a , Z b ] of the model, which underlies the noncommutative gauge theory. We will obtain a non-linear geometric equation for the torsion, which fully captures the underlying Yang-Mills equations of motion of the matrix model. This provides a useful description of the resulting higher-spin gravity on M. The Ricci tensor of the effective metric can then be computed using the equation of motion for the torsion. It turns out that torsion leads to a specific and very interesting modification of the Einstein equations in vacuum here, in contrast to LS or teleparallel gravity. At a more technical level, the present model can be viewed as a reformulation (rather than a reduction) of a 6-dimensional gauge theory to 4-dimensional higher-spin gravity via torsion. The 6-dimensional gauge theory lives on fuzzy CP 1,2 , which is a quantized S 2 bundle over a 3+1-dimensional FLRW space-time M. This is the geometrical description of the underlying background solution as a coadjoint orbit of SO (4,2). The local stabilizer group of a point on M acts non-trivially on the local S 2 fiber, so that the would-be Kaluza Klein modes become higher spin modes, leading to a higher spin theory on M [9,20]. The concept of torsion allows to fully describe this gauge theory in terms of higher-spin gravity. The gauge transformations of the underlying matrix model lead to covariant transformation laws for the metric and torsion, in terms of generalized (higher-spin) Lie derivatives. Although the underlying gauge invariance is exact, its reformulation in terms of 4-dimensional geometry is valid only in an asymptotic regime. Thus gravity and general covariance are understood as emergent phenomena. The main result of this paper is a closed system of non-linear equations for the metric and the torsion in vacuum. Torsion is governed by a Yang-Mills-like equation (5.47) supplemented by a Bianci identity (5.54), and the Einstein equation is obtained with an effective energymomentum tensor due to torsion (5.72). These are equations for higher-spin valued fields 3 . The cosmological background provides an exact solution with torsion but without higher spin components, corresponding to ω = − 1 3 in vacuum. Torsion generally leads to deviations from Ricci-flatness, however this effect is typically small in a weak gravity regime, as the energymomentum tensor for torsion is quadratic. Nevertheless, a rough qualitative estimate suggests that torsion might lead to an apparent "dark matter halo" around very massive objects. Most importantly, the present model has a non-perturbative definition as a Yang-Mills type matrix model, which should make sense at the quantum level. There is indeed intriguing evidence from numerical simulations [21,22] that an expanding 3+1-dimensional space-time structure arises at the non-perturbative level. It should therefore be possible in principle to test, justify and possibly improve the analytical studies with numerical simulations. This paper considers only the vacuum sector of the theory, and the main open issue is how matter acts as a source of torsion and curvature. It is clear that the propagation of matter is governed by the effective metric, and the manifest covariance strongly suggests that matter will lead to the usual source term for the Einstein tensor. However extra derivative terms should be expected, and quantum effects may be important here. This needs to be clarified in future work. The outline of the paper is as follows. We first recall the semi-classical description of the spacetime and the underlying bundle in terms of Poisson manifolds. A mathematical formalism is developed in section 2.1, where the bundle structure is translated into higher-spin valued fields on space-time. Higher-spin valued Lie derivatives are introduced in section 3, which are the key to covariance under higher-spin valued diffeomorphisms. The kinematical setup of the matrix model is translated into this language in section 4, which allows to derive the equation of motion for torsion and the Ricci tensor in sections 5.3 and 5.5. To validate these results, a more pedestrian derivation of the latter is given in section 7.5. The possible role of torsion as dark matter candidate is briefly discussed. Some technical considerations are delegated to the appendix, including the computation of the vacuum geometry in section 7.4.

Matrix model and cosmological spacetime solution
Our starting point is the IKKT or IIB matrix model extended by a mass term, where Z a ∈ End(H), a = 0, ..., 9 are hermitian matrices acting on a Hilbert space H, and Ψ are Majorana-Weyl spinors whose entries are (Grassmann-valued) matrices. Indices are contracted with the SO(9, 1) -invariant tensor η ab . This model has a manifest SO(9, 1) symmetry, and it is invariant under gauge transformations by unitary matrices U ∈ U (H). The model is maximally supersymmetric for m 2 = 0 which is important for its quantization, but we will focus on the bosonic sector here. The only mathematical structures in the matrix model are matrices and commutators, which reduce to functions and Poisson brackets in the semi-classical limit. We must hence learn how to efficiently work with these, and to cast the system into a recognizable geometric form. This depends very much on the background under consideration. Dropping the fermions, the model (2.1) leads to the following equations of motion The basic solution under consideration here is based on special representations of so (4,2). Let M ab be so(4, 2) generators, which satisfy Now consider the doubleton representations H n for n ∈ N, which are minimal unitary highestweight irreps which remain irreducible under SO(4, 1). Then the matrices satisfy the commutation relations (2.6c) It turns out that the operator algebra End(H n ) can be viewed as quantized algebra of functions on a S 2 bundle over some 3+1-dimensional space-time M. To see this, consider the semiclassical limit n → ∞ indicated by ∼, where End(H n ) becomes a commutative algebra C of functions generated by x µ ∼ X µ and t µ ∼ T µ , with Poisson brackets {., .} ∼ −i[., .] arising from the commutation relations. For the doubleton representations under consideration, these generators satisfy additional relations, which in the semi-classical limit reduce to where µ, α = 0, . . . , 3. One can also show that θ µν can be expressed as [9] θ µν = r 2 Then X µ can thus be interpreted as quantized functions onto the region −x µ x µ ≥ R 2 . M turns out to be a two-sheeted cosmological FLRW space-time with manifest SO(3, 1) symmetry. Here η is a global time coordinate defined by which is related to the scale parameter of the universe [12] a(t) 2 = R 2 cosh 2 (η) sinh(η) . (2.11) The sign of η separates the two sheets of M, with a big bounce at η = 0. Similarly, the t µ are extra generators which describe the internal S 2 fiber over every point on M, see figure  2. This S 2 is space-like due to (2.7c) with radius r −2 cosh 2 (η). Together, x µ and t µ generate the algebra C ∼ = C ∞ (CP 1,2 ) of functions on a 6-dimensional bundle, which turns out to be CP 1,2 . This is an SO(4, 1)-equivariant bundle over H 4 , which projects onto M. As explained in [9,23] there is an equivariant quantization map which allows to identify operators with functions up to some cutoff. This paper is devoted to the semi-classical limit, replacing the rhs with the lhs, and commutators by Poisson brackets.
In particular, the relation (2.6b) implies that the derivations act as momentum generators on M, leading to the useful relation {t µ , φ} (2.14) This geometry provides the geometric interpretation of the following solution of (2.3) [9] with all remaining matricesZ a , a = 4, ..., 9 set to zero. This is the cosmic background under consideration. We will consider more general solutions Z µ ∈ End(H n ) of (2.3), which can be viewed as deformations thereof. In the semi-classical limit we can consider these Z µ as elements of the Poisson algebra C using the correspondence (2.12), i.e. as functions on the cosmic background. Any such background defines a matrix d'Alembertian which acts on φ ∈ End(H n ) ∼ C ∞ (CP 1,2 ), and will play a central role in the following.
Higher spin sectors and Poisson brackets. By expanding the algebra C into polynomials of minimal degree s in t µ , we obtain a decomposition into spin s sectors C s . These sectors can also be defined in terms of a so(4, 2) Casimir [9].
Here C 0 are functions of x, and C s φ α (x)t α where t α ≡ t α 1 ...αs . The projection of φ ∈ C to C s will be denoted by [φ] s or φ (s) . The constraints (2.7) reveal that the multiplication respects this grading as follows: In particular, each C s is a C 0 module, so that C can be viewed as space of sections of some vector bundle over M. This is a module of higher-spin fields, and the bundle structure is encoded in the sub-algebra C 0 ⊂ C. The Poisson structure respects this as follows: {t, C s } ∈ C s {x, C s } ∈ C s+1 ⊕ C s−1 (2.19) and the general structure follows from the derivation property: This is a rather complicated structure, but we will argue in the following that typically the terms involving space-time derivatives ∂ are dominant, except possibly in the extreme IR.

Bundle structure and higher-spin fields as push-forwards
The crucial geometrical observation is as follows. The bundle projection Π : CP 1,2 → M (see figure 2) allows to map vectors from CP 1,2 to M via the push-forward, but it does not map vector fields on CP 1,2 to vector fields on M because Π is not injective. However, one can make sense of this as push-forward of vector fields on CP 1,2 to hs-valued vector fields on M, and similarly for tensor products of vector fields. Here hs is the space of functions on the fiber S 2 , which is spanned by polynomials in the t µ . The algebra C of functions on CP 1,2 is viewed as algebra of hs-valued functions on M. This is the crucial concept which will allow to translate the gauge theory on CP 1,2 as a higher-spin gauge theory on M, which can be viewed as generalized gravity theory 4 . Note that the map (2.21) is compatible 5 with SO(4, 1), which implies that the modes in hs translate to higher-spin modes on M.
There are hence 2 different points of view of the present structure. One can take the 6dimensional picture of CP 1,2 as a 6-dimensional manifold, which is a bundle over M. However from the physical point of view, it is better to view this structure in terms of hs -valued functions and tensors on M. We will use both points of view in this paper as appropriate, but emphasize the more physics-oriented 3+1-dimensional picture.
Poisson structure and Hamiltonian vector fields. As usual, the Poisson structure on C allows to associate to any Λ ∈ C a vector field on CP 1,2 , via This is the Lie derivative of φ ∈ C ∞ (CP 1,2 ) along the "Hamiltonian" vector field ξ = {Λ, .}, which defines a (one-parameter family of) diffeomorphism on CP 1,2 , more precisely a symplectomorphism. Restricted to C 0 ⊂ C, this takes the familiar form This is nothing but the push-forward Π * ξ as in (2.21), and we will use the same symbol ξ for both pictures. Clearly the components ξ µ have in general components in different C s sectors, which means that ξ is viewed as hs-valued vector field on M.
Similarly, we can also consider the push-forward of the Poisson structure or bivector field {., .} on CP 1,2 as hs-valued Poisson structure on M, denoted as Note that θ µν = {x µ , x ν } takes indeed values in C 1 , and any coordinates on M can be used here. This respects the Jacobi identity as long as it acts on C 0 , but not in general since the push-forward kills vertical vector fields. The push-forward basically means that hs (or the fiber coordinates) is considered as commutative. We will see that replacing {., .} by {., .} M is a good approximation for functions whose wavelengths are much shorter than cosmic scales and for low spin, since then the Jacobi identity holds to a very good approximation. This will be justified in the next sections. Then ξ = {Λ, .} M defines a hs-valued Hamiltonian vector field on M, and we can drop the subscript M in the appropriate regime. No information is lost in this step, due to the following Lemma: Proof. This follows from the Jacobi identity, since and θ µν together with x µ generate the full algebra C.
The following two sections provide a more detailed justification for a certain approximation, which is used later to obtain the reduced 3+1-dimensional equations. Although this is very important, one may skip these sections at first reading and jump to section 3, where the higher-spin gauge invariance is discussed.

Scales and orders of magnitude
Using SO(3, 1) invariance, we can restrict ourselves to the "reference point" p = (x 0 , 0, 0, 0) on M. Then (2.8) reduces to so that θ 0i ∼ r 2 Rt i θ ij at late times η 1, indicated by ∼. Then the first term in (2.8) dominates, and Note that at p, the t µ generators satisfy and t 0 p = 0. Hence in a sense we are considering a U (su (2))-valued Yang-Mills-type theory of local space-like translations, however the non-commutativity (via the Poisson bracket) leads to novel structures not usually encountered in classical Yang-Mills theory or gravity.
Orders of magnitude and asymptotic regime. Now consider the "size" of the various generators. The relations (2.7) lead to the following scale estimates defining the scale of noncommutativity L NC . This gives for φ (1) where |x| = R cosh(η) measures the cosmic time, and ∼ indicates η 1. Hence the derivative term dominates except for extremely low wavelengths λ, i.e. in the asymptotic regime much shorter than the cosmic scale and longer than the scale of noncommutativity. Similarly, Note that the estimate applies equally to the component in C 0 and C 2 , since θ µν t α and [θ µν t α ] 0 are comparable in size. Hence in general we can write where α is a multi-index. This amounts precisely to the projection to M (2.25) as discussed above, which will be very helpful to extract the leading contributions for physics. Recalling the cosmic FLRW scale parameter (2.11), the ratio of IR and UV scales is at late times. Hence space grows indeed much faster than the NC scale, as it should.

Poisson bracket and reduction to M
To make the reduction to M precise, we derive an explicit formula for the Poisson structure with the form where B and C are antisymmetric. This is not unique due to the constraints An explicit realization is given in appendix 7.2, which leads to the following formula where D and ð are derivations 6 on C given by In particular, DC 0 = 0 means that its push-forward to M vanishes, Π * D = 0. Thus D are "vertical" derivatives along the local S 2 fiber. Moreover, we claim that ð = Π * ð vanishes on the local S 2 fiber. To see that, we compute at the reference point, and note that the first term reproduces the grow of |t| 2 = r −2 cosh 2 (η) = −(rR) −2 |x| 2 in time. Hence the push-forward of a Hamiltonian vector field to M is given by and similarly the push-forward of the Poisson bracket to M is given by which is verified immediately for f, g ∈ C 0 . For late times η 1, we can replace ð by the simpler derivative using (2.27), which agrees with ð at late times. We can then replace (2.41) by This will be used throughout this paper, and we discuss again its range of validity: Asymptotic regime. The fact that D vanishes on C 0 is very helpful to organize the higherspin theory: it leads to the estimate for small spin and η 1, using the abbreviation x · ∂ ≡ R cosh(η)∂. Thus so that we can use the approximation in the asymptotic regime, i.e.
x · ∂f f . If both f and g are in the asymptotic regime (2.31), we can write in agreement with (2.33). Of course we are free to use any coordinates in C 0 here. Thus we have replaced the Poisson structure CP 1,2 by a hs valued bi-vector field on M, which is its push-forward by the bundle projection as discussed in section 2.1. This reduced Poisson tensor will satisfy the following reduced Jacobi identity which holds exactly on C 0 and at least asymptotically for the higher spin components, as long as x · ∂ 1. This in turn implies the identity (cf. Appendix A in [8]) with the same qualifications.
3 Higher-spin gauge invariance and Lie derivatives

Gauge transformation of scalar fields
Consider some scalar field φ ∈ C on CP 1,2 , which transforms under gauge transformations as 7 This is nothing but the Lie derivative of φ ∈ C ∞ (CP 1,2 ) along the Hamiltonian vector field ξ = {Λ, .}. For functions on M i.e. restricted to φ ∈ C 0 , this reduces to which is interpreted as Lie derivative along the hs-valued vector field and we can view ξ µ However one typically cannot get rid of the component ξ (2) = {Λ, x µ } 2 ∈ C 2 , and ξ should be considered as generator of generalized hs-valued diffeomorphisms.
hs-valued scalar fields on M. In the asymptotic regime (2.31), the above formula generalizes to arbitrary φ ∈ C as follows: This is interpreted as Lie derivative of a hs-valued function φ along the hs-valued vector field ξ, indicated byL. This makes sense as long as φ and ξ are in the asymptotic regime.

Gauge transformation of vector fields
Now consider some given Hamiltonian vector field E = {Z, .} on CP 1,2 , such as the frame discussed below. Assume that Z transforms under a gauge transformation (3.1) as Then the associated vector field E transforms as }. This is precisely the Lie derivative of the vector field E along ξ, hence This will give the transformation of the frame in (4.1), and it extends to higher-rank tensor fields as which will apply to the metric tensor. We also recall that the Lie derivative of the Poisson bi-vector field along any Hamiltonian vector field ξ on CP 1,2 vanishes L ξ {., .} = 0 , due to the Jacobi identity.
Gauge transformations and hs-valued Lie derivative on M. Now consider the same from the 4-dimensional point of view. As discussed in section 2.1, the push-forward of the Hamiltonian vector field E = {Z, .} to M for Z ∈ C defines a hs-valued vector field E µ ∂ µ on M, with components Under gauge transformations δ Λ Z = {Λ, Z}, we can still use the first line in (3.8), At this point it is more transparent to use coordinates x µ on M, and the above becomes , using the approximation (2.48) in the second line in the asymptotic regime. This has the standard form of a Lie derivative along a field ξ µ = {Λ, x µ }, and constitutes our "working definition" of a hs-valued Lie derivativeL on M, generalizing (3.6). Remember that both E µ and ξ µ have hs components.
The above considerations provide a 4-dimensional geometrical interpretation of the emergent diffeomorphism invariance and its higher spin extension, which arise 8 from Hamiltonian vector fields on CP 1,2 . Since these preserve the symplectic volume on CP 1,2 , the resulting hs diffeomorphisms on M should be volume-preserving in some sense also from the 4-dimensional point of view. This is indeed the case, as elaborated in Appendix 7.1.

Kinetic action, frame and effective metric on M
We consider the kinetic term for a scalar field φ in the matrix model on a background given by some solution Zα of (2.3) in the semi-classical limit, from two different points of view. which are derivations on C = C ∞ (CP 1,2 ). Frame indices will always be dotted Greek letterṡ α,β, ..., which transform under the global SO(3, 1). This defines a metric as a bi-vector field and the action can be written in the semi-classical limit as 9 where e.g. y A = (x µ , ϑ, ϕ) are coordinates on CP 1,2 . Here the coordinate form of the frame is However, this does not properly reflect the structure of CP 1,2 as S 2 bundle over M, and we will rewrite it in a way which is more transparent from the 4-dimensional point of view.
3+1-dimensional point of view on M. Now consider the reduction (more precisely the push-forward) of the same configuration to M. This applies automatically if φ is in the asymptotic regime. Then is a hs-valued frame on M, which is an invertible C-valued 4 × 4 matrix. Instead of the Cartesian coordinates x µ we could use any other coordinates on M. We can write the volume form (4.5) as where Ω 2 is the volume form of the unit 2-sphere corresponding to the local fiber, and ρ M is the corresponding 4-density. The explicit form is easily obtained as the unique SO(4, 1)-invariant volume on H 4 , cf. [25]. Then the action reduces to absorbing some dimensionful constants in ϕ ∼ φ. Here are an auxiliary and the effective (hs-valued) metric on M, respectively, and is a scalar field (rather than a density). One can also introduce a densitized vielbein which directly gives the effective metric, The last form of (4.8) is indeed hs covariant, in the sense discussed below. This is the dominant contribution for fields φ ∈ C s in the asymptotic regime. The metric G µν indeed governs the d'Alembertian E μ α = sinh(η)δ μ α ,γ µν = sinh 2 (η)η µν , G µν = sinh −3 (η)γ µν = sinh −1 (η)η µν ,ρ 2 = sinh 3 (η) . and the associated vector fields Eα on CP 1,2 transform as using (3.9), where ξ = {Λ, .}. In local coordinates y A , this gives the coordinate expression and we obtain where ξ ν = {Λ, x ν } is a hs-valued vector field, and E μ α is the hs-valued frame (4.6) on M. In particular, this implies where ∇ (γ) is the Levi-Civita connection corresponding to γ µν . We would like to generalize this to the effective metric G µν including the conformal factor. To see this, observe that the invariance of the symplectic form L ξ ω = 0 implies δ Λ ρ 4 ∼L ξ ρ 4 (4.20) for where is the effective volume form (4.7) on M. Thus ρ 2 coincides with the conformal factor for the effective metric (4.9), and we obtain for the effective metric in the asymptotic regime. Here ∇ (G) is the Levi-Civita connection corresponding to G µν . To summarize, the gauge invariance arising from symplectomorphisms on CP 1,2 leads to an emergent higher-spin symmetry in 4 dimensions. This is achieved by considering E μ α , G µν and ξ µ as hs-valued vector fields on M 3,1 , and (4.23) should be understood in this higher spin sense. In the linearized regime for s = 1, this reduces to the standard formulas for volume-preserving diffeomorphisms in 4 dimensions [12].

Geometric description of the non-linear regime
In this section, we develop a geometric formalism based on a higher-spin generalization of the Weitzenböck connection and torsion. We will work in the asymptotic regime (2.31), using the 4-dimensional point of view developed above.

Weitzenböck connection and torsion
The fundamental degrees of freedom of the matrix model is the background Zα and its associated vielbein Eα = {Zα, .}. It is then natural to define a connection which respects the vielbein, analogous to the Weitzenböck connection [26], along Eγ. This can always be solved as provided E ρ α is an invertible matrix taking values in C. In the perturbative regime, the most significant contribution should be the C 0 components of E μ α and Γ μ γρ , accompanied by some higher-spin contributions. In local coordinates y µ , this is where we define the derivation 10 The inverse vierbein is defined as usual This connection is automatically compatible with the metric γ µν , For any hs-valued vector field V µ on M, we can then define the covariant derivative as This connection is flat 11 since the frame is parallel, ∇Eβ = 0. However it typically has torsion, using the Jacobi identity. Due to Lemma 2.1, T μ αβ fully captures the noncommutative field strengthΘαβ. Thus torsion encodes the quantum structure of space-time, and it is the semiclassical shadow of it. This is the key to understand the matrix model in terms of gravity 12 , and the equations of motion of the model will be re-formulated in terms of torsion below. For a perturbed cosmic background Zα = tα + Aα, we havê so that thatΘαβ ∈ C 1 up to higher-spin corrections. More explicitly, the torsion tensor is The gauge transformation of the torsion tensor in the asymptotic regime is obtained again from (3.13), where ξ ν = {Λ, x ν } is a hs-valued vector field on M. Together with (4.18) this implies δ Λ T µ ρσ ∼L ξ T µ ρσ (5.13) and similarly for the effective frame using (4.20). Hence torsion transforms as a covariant tensor, just like the metric.
Relation with the effective Levi-Civita connection. Now consider the Levi-Civita connection ∇ (γ) for the metric γ µν , which is obtained as usual from the Christoffel symbols (5.14) 12 The role of torsion in a dimensionally reduced noncommutative gauge theory as related to gravity was already pointed out in [17], however the specifics are different. Torsion arises in (5.9) as derivative of the NC field strength, unlike in [17]. Also, the action in previous work is typically given by a contraction of torsion, which is not the case here. See also e.g. [27] for other work related to torsion in a similar context.

Here
is (a higher-spin analog of) the contorsion of the basic Weitzenböck connection, which is antisymmetric in νσ. Therefore Similarly, the Levi-Civita connection ∇ (G) for the effective metric G µν is obtained as which together with the above gives will be recognized below as Weitzenböck connection and contorsion of the effective frame, and accordingly the indices should be raised and lowered with G µν . To avoid any confusion with the two metrics γ µν and G µν , all connection and (con)torsion symbols will be written with two lower and one upper index, where no ambiguity arises. This allows to rewrite the effective Levi-Civita connection in terms of the Weitzenböck connection and the torsion, which amounts to the simple rule for the covariant derivatives and similarly for higher-rank tensors.
Effective (rescaled) frame. The rescaled or effective frame (4.11) for the effective metric 13 gives rise to an associated Weitzenböck connection which is compatible with the effective metric,∇ E μ α = 0 =∇G µν . (5.22) 13 Note that E μ α is not the Hamiltonian vector field associated to 1 ρ Eα.
One must be very careful with the frame indices, since there are two different frames in the game. It is therefore safer to use the coordinate form. Theñ For the covariant derivatives, this amounts to the simple rulẽ and similarly for higher-rank tensors. Then the torsion tensor is and the effective contorsion is related to that of the basic frame as follows in complete agreement with (5.19). Calligraphic fonts (or a tilde) indicate the rescaled frame.
(Harmonic) normal coordinates. We will denote coordinates around some point p ∈ M as normal coordinates at p if the Christoffel symbols vanish at p, Hence if torsion vanishes, the Levi-Civita connection coincides with the Weitzenböck connection, and both are flat. This can be done either for the basic metric γ µν or for the effective metric G µν , but typically not for both simultaneously. In the present context, the Christoffel symbols will be hs-valued in general, and so are 14 the normal coordinates y µ . In particular, consider harmonic coordinates local coordinates y µ on M around p, which by definition satisfy G y µ = 0. We can demand in addition that at any given point they are also normal coordinates, so that To see that they exist, it suffices to note that one can find harmonic functions y µ with any prescribed "boundary value" and normal derivative for any hypersurface through 'p; thus one can choose the Γ (G) µ νσ | p freely up to Γ µ (G) , which vanishes by the harmonic condition. We denote such coordinates as "harmonic normal coordinates", which will be used in the alternative derivation of the Ricci tensor in section 7.5.

Some useful identities
It was shown in [8] that for symplectic manifolds, the Matrix or Poisson d'Alembertian (2.16) is proportional to the metric d'Alembertian The same relation holds in the present reduced 4-dimensional setting 15 : in the asymptotic regime 16 , with ρ given in (4.10). Furthermore, the following identities hold This is proved in appendix 7.3, by writing out using vielbein and Weitzenböck connection and using (2.51). This is a generalization of a similar result in [8] for symplectic branes. It leads to the following explicit formulas for the contraction of the (con)torsion: Lemma 5.2. The (con)torsion associated to the basic frame satisfies For the rescaled frame resp. effective metric, we have Proof. Using the above results, we can evaluate the contraction of the torsion using |γ µν | = ρ 2 ρ −1 M , and using (5.15). For the rescaled frame, (5.23) gives

Equation of motion for the torsion
The second term can be rewritten as using the approximation (2.48) in the asymptotic regime. Furthermore, we note that since ∇E = 0. Combining this we obtain which amounts to Upon lowering an index with γ µν we obtain This non-linear equation encodes the non-linear structure of the Yang-Mills equations of motion (5.41). Contracting with γ µρ gives The first term can be evaluated using (5.35) and (5.31) and (5.33) as We therefore obtain According to the discussion in section 2.3, all these equations are exact for C 0 , and asymptotic for the higher spin components. We verify in appendix 7.4 that these equations are indeed satisfied exactly for the background solution and its torsion. If desired, (5.47) can be rewritten in terms of the Levi-Civita connection using (5.20), It could be expressed in terms of the torsion only, but this does not lead to a simpler expression. Together with the equation (5.72) for the Ricci tensor and the Bianci identity below, this provides a closed system of equations for the metric and the torsion.

Bianci identity
There is a Bianci-type identity for the torsion, which results from the Jacobi identity where (cycl) indicates cyclic permutations inα,β,γ. Its coordinate version is using again (2.48), or equivalently which is cyclic in λ, ρ, σ. Together with (5.47) we obtained an analog of the Yang-Mills equations. Contracting σµ, this gives using Lemma 5.2. The middle terms can be rewritten using and we obtain the identity Contracting (5.54) with γ σλ does not give any non-trivial relation.

Vacuum equation for the Ricci tensor
Now we compute the Ricci tensor for the Levi-Civita connection associated with the effective metric G µν . This is achieved by expressing the Riemann tensor in terms of the torsion. We start from In (Riemann) normal coordinates at p ∈ M, this simplifies using (5.18) as using (5.28). It remains to evaluate the derivative terms of the contorsion. The first term can be evaluated using (5.19), which gives where ∂ρ · ∂ρ := G µσ ∂ µ ρ∂ σ ρ .
The first term can be evaluated as using (5.20), (5.19) and (5.36). This gives after some straightforward algebra using (5.26) So far this is an identity. We now replace the contorsion with the torsion using (5.15) and use the Bianci identity (5.57) as well as the equation of motion (5.47) for the torsion. This gives and we obtain the desired equation for the Ricci tensor of the effective metric in vacuum This agrees precisely with the result (7.51) obtained in a more pedestrian way. The last term can be rewritten in terms of the torsion using the contracted equations of motion (5.50), This is an algebraic equation for the Ricci tensor in terms of torsion 17 . The Ricci scalar is obtained by contracting with G νσ , T µσρ T µσρ + 2ρ −2 ∂ρ · ∂ρ + 2m 2 (5.71) using (7.56) in the last step. Hence the present vacuum equations can be written as Einstein equations in the form with an effective energy-momentum tensor due to the torsion, recalling that m 2 = 3R −2 . This is verified for the cosmic background in section (7.4). The conservation law ∇ ν (G) T νµ = 0 is guaranteed at least in vacuum because the rhs simply computes the Einstein tensor, which is conserved. The above equations (5.47), (5.54) and (5.72) for the torsion and Ricci tensor provide a closed system of non-linear equations which govern the emergent gravity on the present background. The fact that the quantities are hs-valued makes them rich and rather complicated. Although the present derivation is restricted to the asymptotic regime for the hs sector, the equations are exact for the lowest components in C 0 , as verified for the cosmological background. Using covariance under higher-spin gauge transformations, it should be possible to largely transform away the hs-components in many situations, and the exact equations for the C 0 component should be accessible to analytic investigation.

Discussion and further considerations
The crucial point of the above result is that T µν = O(T T ) is quadratic in the torsion, as appropriate for an energy-momentum tensor. In contrast, the Riemann tensor (5.61) contains a (derivative) term which is linear in the torsion, hence any non-trivial geometry has nontrivial torsion. However, this linear contribution vanishes on-shell for the Ricci tensor, so that vacuum geometries are Ricci-flat up to higher-order (non-linear) contributions. This means that the present theory is a serious candidate for gravity, and deviations from GR (at least in vacuum) are restricted to the non-linear regime. It remains to quantify the deviations of the present theory from GR. One regime where deviations will surely arise is the strong gravity regime, where the Riemann tensor is large in some sense. That is the regime where deviations from GR typically arise in alternative approaches to gravity such as string theory. However there is another regime where torsion may lead to significant modifications here, namely for very large, massive objects as discussed in the next section. This new mechanism arises from the self-coupling of torsion due to (5.47), and it might mimic the presence of dark matter as discussed below.
Torsion as a "dark matter". Consider first the linearization of the torsion around the cosmic background torsionT = O( 1 a(t) ) (7.26). Since the contributions from the background cancel, the eom (5.51) takes the following schematic form The constant and linear terms on the rhs are suppressed by the cosmic scale factor 1 a(t) . However, since the torsion is smaller than the cosmic background in the linearized regime, it will not be very significant physically. Therefore we need to consider the non-linear regime, To get a rough qualitative idea what this could mean, consider the simplified radial equation for t = t(r), where r is the distance to the center of some massive object. This should give a reasonable picture for the radial dependence of the torsion tensor. That equation has the general solution for some constant c. This leads to an effective energy-momentum tensor 8πT µν ∼ 1 (r+c) 2 (5.73) as a source of the Einstein tensor, which would behave like a dark matter halo with density profile ρ DM (r) ∼ 1 (r+c) 2 corresponding to a total mass leading to a rotation velocity v rot (r) which is roughly independent of the distance r to the (galactic) center. This is indeed what is typically observed. The scale parameter c should be determined by continuity in a refined treatment, and it is presumably set by the "size" or mass of the object. At very large distances, the torsion (5.78) will merge to that of the cosmic background, leading to a natural cutoff for the effect. For small masses or objects, there will not be sufficient space for (5.78) to rise significantly above the background, so that the effect should be significant only for very large objects such as galaxies.
Needless to say that this crude qualitative consideration needs to be considerably refined before quantitative statements can be made, and the coupling to matter needs to be understood and taken into account properly. In any case, it is intriguing to obtain a qualitatively reasonable first estimate, and it is also encouraging that the present mechanism based on dynamical torsion is sufficiently rich that different types of behavior might be produced.
Coupling to matter. This paper is restricted to the vacuum geometry of the model. To properly talk about gravity we should of course take matter into account, which is indeed an intrinsic part of the IKKT matrix model. It is clear that the kinematics of matter is properly governed by the metric; this is how the metric was identified 18 . However, the non-trivial question is how matter acts as a source for torsion and the Ricci tensor. It is not clear what is the best way to work this out, and we postpone this question to future work. However, a few comments can be made at this point: First of all, due to the (higher-spin) covariance of the theory it is highly plausible that the energy-momentum tensor for matter will arise on the rhs of the Einstein equations. However there will also be higher-derivative terms, and the question is if the standard contribution dominates the higher-derivative contributions; see also the related discussion in [9]. Covariance will strongly restrict the possible terms, and it is certainly plausible that the energy-momentum tensor will dominate. Due to the presence of several scales on the background this must be studied in detail, in order to identify the effective Newton constant. Since quantum effects are typically significant in this context, this may not be a trivial task. Similarly, the effect of matter on torsion must be understood. This is expected to be small since for bosonic matter there should not source torsion at all (as only the metric appears in the kinetic term), and for fermions the effect is expected to be small as well, due to the supersymmetry of the underlying matrix model.

Conclusion and outlook
The present paper provides a tensorial description of the vacuum sector of the effective gravity which arises on a solution of the IKKT matrix models found in [9], interpreted as FLRW spacetime. The noncommutative Yang-Mills gauge theory is cast into a geometric form which makes the gauge invariance manifest, in terms of a higher-spin generalization of volumepreserving diffeomorphisms. The crucial concept turns out to be torsion, or rather a higherspin generalization of torsion, which encodes the quantum structure of space-time and provides its semi-classical shadow.
Torsion turns out to be an independent and additional physical quantity besides the metric, and the Einstein equations for vacuum are modified through an effective effective energymomentum tensor due to torsion. A non-linear equation for torsion is obtained, which encodes the underlying Yang-Mills-type equations of motion of the matrix model. This equation is exact for the standard (lowest-spin) tensorial components, but obtained only in an intermediate ("asymptotic") regime for the higher-spin components. Moreover, we have argued that at least in the vacuum sector, the modification of GR due to torsion should be small except for very large objects such as galaxies, and on cosmic scales. The point is that torsion enters quadratically in the effective energy-momentum tensor, while it is governed itself by some non-linear PDE. A rough qualitative estimate suggests that it could indeed behave like an apparent dark matter halo around galaxies. In principle, the equations obtained in this paper should allow to obtain a quantitative description for this effect which can be tested. The theory of gravity obtained in this way is governed by an action which is very different from the Einstein-Hilbert action. Unlike in the teleparallel formulation of general relativity, there is no way to rewrite the matrix model as local action in terms of the torsion, metric and frame. It is precisely this non-standard non-geometric origin which makes the present approach to gravity so interesting and potentially far-reaching. However, the present paper is limited to the vacuum sector of gravity. This restriction is only due to technical reasons, and obviously needs to be closed in future work. In principle, everything should follow from the underlying matrix model, and matter will certainly influence the geometry in some way consistent with the (generalized) covariance. Moreover, the framework of matrix models allows to make sense of the path integral. In partial for the maximally supersymmetric IKKT model, one may reasonably hope that the present (semi-) classical treatment is not too far from the full quantum theory. The extra structure required for an interesting matter sector can naturally arise from fuzzy extra dimensions realized by the extra 6 bosonic matrices in the model, as discussed e.g. in [28][29][30][31], see also [32]. Finally, a general message is that we ought to be cautious in extrapolating general relativity to regimes where it was not directly tested. The present theory may reproduce GR quite well in intermediate regimes and it has a healthy linear excitation spectrum without ghosts [10], but it certainly differs significantly on very long scales. Thus the correct theory of gravity may be far richer than GR, and the puzzles of dark matter, dark energy and the cosmological constant may well be evidence supporting such a picture.

Volume-preserving diffeomorphisms
Consider the vector field (3.3) associated a gauge transformation generated by Λ (s) ∈ C s Using the basic identities (A.35) in [9], it is easy to see that these components satisfy the following constraint Here∇ is the Levi-Civita derivative w.r.t. the cosmic background metric [12]. In this sense, ξ µ can be interpreted as volume-preserving higher spin diffeomorphism.

Calculations for the Poisson bracket
We claim that the following formula realizes the ansatz (2.35) for the Poisson brackets: The two terms look different due to the ambiguity in (2.35), and different forms can be obtained using This leads to the following closed formulas for the derivatives ð and D in (2.38): For the generators, this gives One can then verify (7.4) and (2.37) explicitly.
Locally rescaled generators. To get a better intuition for the Poisson brackets, let us define adapted momentum generators for some cosmic time scale η 0 near some observer, where β 0 = cosh −1 (η 0 ) 1. They satisfy approximately canonical commutation relations The noncommutativity of the remaining generators is obtained from (2.26), at the reference point. Hence

Proof of Lemma 5.1
Consider first the following contraction This gives (5.34) which has the same form as for the Christoffel symbols for the Levi-Civita connection. Now consider the d'Alembertian φ: Alternatively, we can proceed as In particular, this gives (5.33) As a check, the second line in (5.14) reproduces the standard identity in Riemannian geometry To show (5.32), we observe that the basic frame can be written using (2.41) as follows using the identity ∂ µ (ρ M θ µσ ) = 0 (2.51) which follows from the Jacobi identity. Together with ∂ µ E μ β = −Γ µ µβ this gives which gives (5.32). We can use this to continue (7.13) as follows