Emergent gravity on covariant quantum spaces in the IKKT model

We study perturbations of 4-dimensional fuzzy spheres as backgrounds in the IKKT or IIB matrix model. Gauge fields and metric fluctuations are identified among the excitation modes with lowest spin, supplemented by a tower of higher-spin fields. They arise from an internal structure which can be viewed as a twisted bundle over $S^4$, leading to a covariant noncommutative geometry. The linearized 4-dimensional Einstein equations are obtained from the classical matrix model action under certain conditions, modified by an IR cutoff. Some one-loop contributions to the effective action are computed using the formalism of string states.


Introduction
Matrix models such as the IIB or IKKT model [1] (cf. [2,3]) provide fascinating candidates for a quantum theory of fundamental interactions. Their most interesting feature is that geometry is not an input, but arises itself as a brane-type solution with dynamical "quantum" geometry. Fluctuations around such solutions lead to gauge fields and matter fields on the background. It is natural to expect that gravity, along with the other fundamental interactions, should emerge on suitable backgrounds in the low-energy, semi-classical regime. Remarkably, numerical evidence for the emergence of 3+1-dimensional space-time within the finite-dimensional IIB model was reported recently [4,5].
At first sight, the relation of the IIB matrix model with string theory suggests that 4dimensional gravity can arise only if target space is compactified. This would not only lead to the well-known issues with a vast landscape of possibilities, it would also require ad-hoc modifications or constraints 2 of the matrix model, destroying much of its appeal and simplicity. With this motivation, there were ongoing efforts to understand possible mechanisms for gravity in this model based solely on the 4-dimensional, non-commutative (NC) physics of the branes rather than the 10-dimensional bulk gravity (which arises in the matrix model upon quantization) [7][8][9][10]. Although 4-dimensional NC gauge theory behaves indeed very much like a gravitational theory [8,9,[11][12][13], the emerging gravity on basic branes seems to be different from usual gravity, and it was not possible to derive the Einstein equations up to now.
In this paper, we show that Einstein-like gravity can indeed arise on more sophisticated, covariant noncommutative branes in this model, at least in some regime. This is based solely on the classical matrix model dynamics for fluctuation modes on the background brane, and has nothing to do with IIB supergravity in the bulk. The internal structure of the quantum space is crucial for the mechanism. This background is a generalized 4dimensional fuzzy sphere S 4 Λ , but most of the considerations should apply also to analogous spaces with Minkowski signature.
There are two crucial features of S 4 Λ which are essential here [14]. First, it has an internal bundle structure, which transforms non-trivially under local space(time) rotations. Each point on the local fiber corresponds to a particular choice of an antisymmetric tensor θ µν on S 4 . This tensor is averaged over the fiber, leading to a covariant noncommutative structure of the 4-dimensional space. The second crucial feature is the fact that θ µν (complemented by P µ ) is not central, but generate the local Euclidean isometry group including translations. Quantum spaces with these features will be denoted as covariant quantum spaces. This concept is actually very old and goes back to Snyder and his proposal [15] for a Lorentz-invariant noncommutative Minkowski space. Fuzzy S 4 is a compact and wellcontrolled Euclidean version of such a space. Due to the extra generators θ µν and P µ , the corresponding algebra (of "functions") is larger than what seems to be needed in field theory, hence this type of space was not very much appreciated.
In contrast to most previous work on this type of spaces (cf. [16,17] and references therein), we take serious these extra fluctuation modes. They can be understood as harmonics on the internal fiber, which -in contrast to Kaluza-Klein compactification -transform nontrivially under the local isometry group. This leads to an infinite tower of higher-spin fields, truncated at N for fuzzy S 4 N . Among the lowest modes in this tower, we identify the metric fluctuation, a selfdual SO(4) connection as well as gauge fields. We then perform a fluctuation analysis in the semi-classical limit along the lines of [7,8]. The metric fluctuation H µν is a combination of a rank 2 tensor field h µν and the divergence ∂ ρ A µρν of a SO(4)valued gauge field. Their semi-classical equations of motion of the classical matrix model then lead to the (linearized) Einstein equations for H µν , for length scales below some IR cutoff scale m −1 . Above this length scale, gravity no longer applies. However, this requires a certain type of generalized fuzzy spheres S 4 Λ , and we have to assume dimensional reduction to 4 dimensions. Mechanisms to ensure this are suggested, but this needs to be addressed in future work. This issue could be avoided by a suitable self-dual modification of the matrix model action.
The present framework incorporates several aspects of previous work in this context. Averaging over the Poisson structure θ µν was considered in the DFR approach to field theory on the Moyal-Weyl plane [18], in order to preserve Lorentz invariance. However, θ µν was considered as central there, which kills gravity. The effective metric and the dynamics of NC branes in matrix models was analyzed in [7,8], but the backgrounds under consideration were too simple. Finally, an interpretation of the matrices as covariant derivatives rather than position operators was proposed in [19]. This also leads to higher spin fields with some similarities to the present framework and even the Einstein equations in vacuum, however this doesn't work in the finite-dimensional model, and the proper coupling to matter was not established. Due to the SO(5) setup, there are also similarities with the MacDowell-Mansouri formulation of GR [20,21], however the physics is different: the full SO(5) symmetry is manifest here, and there are additional degrees of freedom beyond the ones in GR. The present framework shares aspects with noncommutative SO(5) gauge theory approaches [22][23][24], but again this is not quite appropriate: the gauge group is actually much larger here, corresponding to (a quotient of) U (so (5)).
The reason for insisting on the IIB model is that the quantization is well-behaved, since the non-local UV/IR mixing is mild due to maximal SUSY. In section 6, we compute the leading terms in the one-loop effective action for the lowest fluctuation modes on S 4 N . This is possible due to recent progress for the quantization of field theory on fuzzy spaces based on string states [25]. We show that previous one-loop results can be reproduced efficiently in this formalism, and some (preliminary) computations suggest that the one-loop effects can be captured by a minor generalization of the classical action, preserving the mechanism for gravity. This paper is written in a pedestrian way, to make everything explicit and to avoid getting trapped in some formalism. Of course there should be a more structural approach, and many limitations of this paper -notably the restriction to the linearized regime -are clearly inessential. Other open issues include the coupling of the conformal mode to scalar fields which seems odd (see section 4.2), the proper extension to the Minkowski case, the justification of dimensional reduction for the generalized sphere, and the coupling to fermions. These should be addressed in future work. Nevertheless the basic mechanism is compelling, and should provide a serious candidate for a quantum theory of gravity which behaves similar to GR in a suitable range.
2 Covariant fuzzy four-spheres S 4 Λ We consider covariant fuzzy four-spheres defined in terms of 5 hermitian matrices X a , a = 1, ..., 5 acting on some finite-dimensional Hilbert space H, which transform as vectors under SO (5) [M ab , X c ] = i(δ ac X b − δ bc X a ), [M ab , M cd ] = i(δ ac M bd − δ ad M bc − δ bc M ad + δ bd M ac ) . (2.1) Here the M ab = −M ba for 1 ≤ a = b ≤ 5 define a (not necessarily irreducible) representation of so(5) on H. The radius is a scalar operator of dimension L 2 , and the commutator of the X a will be denoted by Here and throughout this paper, indices are raised and lowered with g ab = δ ab . This type of relations constitute a covariant quantum space.
The form of the algebra (2.1) suggests a particular realization of such fuzzy four-spheres, based on an irreducible representation (irrep) of so(6) as follows 3 Here M ab , a = 1, ..., 6 define an irrep of so(6) ∼ = su(4) on H, and r is a scale parameter of dimension L. Correspondingly, so(5) ⊂ so(6) is embedded by restricting the indices of M ab to a, b = 1, ..., 5. We also note the following simple identity for such spheres This is the type of space under consideration in this paper. There are important differences depending on the representation H of so (6): The basic fuzzy 4-sphere S 4 N . The simplest example is the "basic" fuzzy four-sphere S 4 N [27][28][29], which is obtained for the highest weight irrep H = H Λ of so(6) with Λ = (0, 0, N ), denoting highest weights by their Dynkin indices. This representation can be realized as totally symmetric tensor product H Λ ∼ = (C 4 ) ⊗ S N of the 4-dimensional (spinor) representation of so (6), which happens to remain irreducible as a so(5) ⊂ so(6) representation. In this particular case, the radius operator is proportional to the identity operator, For this basic fuzzy sphere S 4 N , the following useful formulae are established in appendix B where {., .} + denotes the anti-commutator. As explained in appendix A, this is the quantization of a 6-dimensional coadjoint orbits of SO(6) mapped to S 4 → R 5 via the x a ∼ X a .
Generalized fuzzy 4-spheres S 4 Λ . More general fuzzy 4-spheres are obtained for so(6) irreps H Λ with Λ = (n 1 , n 2 , N ). As explained in appendix A, these arise as quantizations of generic coadjoint SO(6) orbits, and the semi-classical geometry is that of a "thick" 4-sphere embedded in R 5 . As long as n 1 , n 2 N , the radius R 2 = X a X a is non-trivial but with sharply peaked spectrum around As explained in Appendix A, S 4 Λ can be understood as a SU (3) bundle over S 4 N . The relation (2.7) is modified as where t ab = O n N , see (B.5) and (A.19). This generalization will be essential for gravity.

Semi-classical geometry and mode decomposition
As usual for fuzzy or noncommutative spaces, the matrix algebra End(H) constitutes the noncommutative algebra of functions or fields on the (generalized) fuzzy 4-sphere.
The realization of M ab in terms of generators of so(6) ∼ = su(4) also provides the proper geometrical interpretation. We recall the well-known fact that End(H Λ ) can be naturally interpreted as quantized algebra of functions on the coadjoint orbit O[Λ] = {g · Λ · g −1 ; g ∈ SU (4)} of su(4) through the weight Λ (cf. [30]). The generators M ab , a, b = 1, ..., 6 are quantized embedding functions dual to some ON basis λ ab of su(4). In particular, the X a ∼ M a6 are naturally interpreted as projections of such coadjoint orbits to S 4 ⊂ R 5 , where Π denotes the projection of su(4) to the subspace spanned by the λ a6 generators. Hence the fuzzy 4-spheres are actually higher-dimensional homogeneous spaces which are twisted bundles over S 4 , with the fiber playing the role of a hidden extra dimension. In contrast to standard Kaluza-Klein compactifications, these extra dimensions lead to higherspin modes here. For the basic 4-sphere S 4 N , the underlying orbit is O[Λ] = CP 3 , which is a S 2 bundle over S 4 as elaborated in [14,[31][32][33][34][35][36]. More details on the geometry including the generic case are given in Appendix A. In particular, the space of classical functions on these orbits is spanned by polynomials F (x a , m ab ), which are in one-to-one correspondence with the noncommutative modes (2.26), up to some UV cutoff defined by N and n i .
whose projection (push-forward) to S 4 is given by (2.14) Here ξ are coordinates on the internal fiber of O[Λ] over S 4 . For the basic 4-sphere S 4 N , θ µν (x, ξ) is selfdual (SD) at each (x, ξ), defining a bundle of SD frames over S 4 , which rotates (and averages out) along the fiber S 2 . More precisely, it transforms as (1, 0) under the local SO(4) = SU (2) L ⊗ SU (2) R rotations, which are implemented by the action of {θ µν , .} on itself. In the noncommutative case, this amounts to a gauge transformation In other words, local rotations are implemented as gauge transformations, which already hints towards gravity.
For the 5 embedding functions x a ∼ X a , the Poisson bracket x a = x|X a |x ≡ X a . (2.17) Up to a U (1) phase factor, they are in one-to-one correspondence to points x on O[Λ]. Alternatively, we can use the SO(5) point of view and consider highest weight states of the SO(5) modules. It will suffice here to consider the case of the basic fuzzy sphere S 4 N , where both notions coincide. We can then label the points on the bundle O[Λ] ∼ = CP 3 locally by x ∈ S 4 and ξ, where the "north pole" p corresponds to the highest weight state |Λ . They are optimally localized, minimizing the uncertainty in position space 5 which defines the length scale L N C . One can then associate to any operator φ ∈ End(H) This also encodes the uncertainty scale L N C and the volume quantization via (2.8 This locally separates into an integration over S 4 times the internal fiber F, which allows to evaluate the matrix model actions in a standard semi-classical form. Scalar fields and higher-spin modes. The most general functions on fuzzy S 4 N are organized into the following SO(6) resp. SO(5) modes (cf. [29,31,35,36]) denoting highest weight irreducible representations (irreps) by their Dynkin indices; for example, (n, m) denotes the so(5) irrep with highest weight Λ = nΛ 1 + mΛ 2 where α 1 is the long root and α 2 the short root. We are mainly interested in the "low spin" representations with small m. Then a more explicit realization is obtained in terms of ordered polynomials 6 in the generators X a and M ab , a, b = 1, ..., 5. For example, scalar fields on S 4 correspond 5 Here p|X 5 |p = r 2 N at the north pole p follows using the explicit realization of X a in terms of gamma matrices [28]. 6 This is a quotient of the Poincare-Birkhoff-Witt basis of U (so (6)).
to the (n, 0) modes, realized by totally symmetric polynomials F (X) = F a 1 ...an X a 1 ...X an , and denoted by is constant along the fiber and defines a function on S 4 . There is an associated projection map [29] φ → [φ] 0 := φ 0 ∈ C N (S 4 ), (2.25) which picks out the scalar modes (n, 0) in (2.22). In the semi-classical limit, this corresponds to integrating φ(x) over the internal fiber.
More generally, we can organize all other higher spin fields in terms of polynomials with "internal" generators M ab , a, b = 1, ..., 5 multiplied by scalar functions. For example, ..an;bc X a 1 ...X an M bc ∈ (n + 1, 2) so(5) ..an;bc;de X a 1 ...X an M bc M de ∈ (n + 2, 4) so(5) (2.26) and so forth, where the F a 1 ...an;bc and F a 1 ...an;bc;de are tensors of SO(5) corresponding to Young tableaux with one row of length 2 and two rows of length 2, respectively. In particular, the F bc (X)M bc can be identified with 2-forms F bc (x)dx b ∧ dx c on R 5 . These (n, m) modes with m = 0 correspond to functions on O[Λ] ∼ = CP 3 which are non-trivial harmonics on the S 2 fiber. They are higher-spin fields on S 4 rather than Kaluza-Klein modes, because the local Lorentz group acts non-trivially on the internal S 2 fiber. This leads to a higher-spin theory, and we will show that its spin 2 sector describes gravity, but only for the generic spheres S 4 Λ . For the generalized spheres S 4 Λ , the scalar operator R 2 = X a X a is a non-trivial so(5) Casimir operator which distinguishes some of the internal structure. Then the mode decomposition is analogous but more complicated, with multiplicities arising in the decomposition (2.22). E.g. for Λ = (k, 0, N ), one finds schematically (2.27)

Local description
We would like to understand the local structure from a field theory point of view, near some reference point p ∈ S 4 denoted as "north pole". We pick a coherent state |p to mark this point. Throughout this paper, tensorial objects at the (arbitrary) point p ∈ S 4 will be expressed in terms of the local tangent space T p S 4 , using the 4 tangential Cartesian coordinates x µ centered at p, with x µ (p) = p|X µ |p = 0, assuming n i N . Then quantities such as x µ (p) can always be dropped, greatly simplifying the analysis. Thus we can view x µ as Riemannian normal coordinates at p with respect to the embedding metric g µν of S 4 ⊂ R 5 , and ∇ [g] µ | p = ∂ µ | p . To avoid confusions with the effective gravitational metric, we will use the symbol ∂ µ for ∇ [g] µ , and we will often drop its ("cosmologically small") curvature [∇ for simplicity. The generators separate accordingly as with X 5 = R 2 − X µ X µ , and the 4 matrices X µ ∼ x µ are quantizations of these local coordinates. The stabilizer of p (or X 5 ) is given by SO(4). Accordingly, so(5) decomposes into so(4) and local "translation" generators, In this setup, the (Euclidean) Poincare-group ISO(4) is recovered as usual by a contraction taking R to be much larger than any other length scale under consideration. Then the S 4 Λ algebra takes the form assuming r 1. Here and in the following, greek indices indicate that the corresponding tensor is tangential.
Poisson algebra limit. Now consider again the semi-classical (Poisson) limit. The exact relations (2.7) for S 4 N then imply the following important formula where P ab T is the projector on the tangent space of S 4 ⊂ R 5 . This allows to evaluate the kinetic term of a scalar field φ ∈ C N (S 4 ) (2.23) in the semi-classical limit: (2.34) As always, this is obtained by replacing commutators with Poisson brackets. Here will play a prominent role as effective background metric. In contrast to θ µν , this is indeed a tensor on S 4 , i.e. it is constant along the fiber. For the translation generators p µ = 1 Rr 2 θ µ5 , equation (2.8) implies This shows that on S 4 N , the functions p µ are not independent and basically vanish, since x µ = 0 in the local frame at any given point p ∈ S 4 . See appendix A for more details.
In contrast for the generic spheres S 4 Λ , the p ν are independent functions, and (2.33) is replaced by where t ab has a non-vanishing radial component. Upon averaging over the local fiber, this defines a scale L R characterizes the thickness of the sphere S 4 Λ with Λ = (n 1 , n 2 , N ), and the contribution N in (2.38) arises from the uncertainty 7 of X µ in (2.8). In particular, this gives

Functions versus symmetry generators
It is important to keep in mind the double meaning of the generators θ µν and P µ : 1. as symmetry generators of the isometry group, which act on wavefunctions via the adjoint. Then the normalization M µν = r −2 Θ µν is appropriate. In particular P µ and M µν generate the local Poincare algebra for R → ∞.

as generators of the algebra of functions on O[Λ]
(along with x µ ), viewed as bundles over S 4 . In the fuzzy case, this is replaced by End(H Λ ), which describes the degrees of freedom in a field theory (or matrix model) on S 4 Λ . This algebra is "almost" commutative for large Λ. 7 This can also be seen using e.g. the explicit representation of S 4 N on H = (0, N ). Similar effects are well-known for S 2 N .
Consider e.g. the basic sphere S 4 N . Since the underlying space O[Λ] ∼ = CP 3 is 6-dimensional, there are locally only 6 independent coordinate functions. At the north pole, these are the x µ , plus 2 of the 3 selfdual θ µν variables which parametrize the fiber S 2 . The θ µν can be viewed as function on CP 3 taking values in the SD 2-forms Ω 2 (S 4 ) or in the local su(2) L ⊂ so(4). However, the p µ functions vanish in the semi-classical limit, as explained above. Therefore there are no modes of the type F µ (X)P µ on S 4 N , they only exist on the generalized spheres S 4 Λ ; this will be crucial below. Now consider the symmetry generators and their action on wavefunctions. The quadratic Casimir of SO(5) can be written as using the same symbols for the abstract generators as in the S 4 N algebra. Acting on scalar fields (or on fields with low spin), the angular momentum contribution can be neglected 8 compared with the translational contribution, C 2 [so(5)] ≈ R 2 P µ P µ . Comparing the formula for its eigenvalues for the low-spin modes m = 0, 1, 2. A simpler way to understand this is via the semiclassical form of the free action for scalar fields [8] T where Ω is the symplectic volume form, in agreement with (2.44). This shows again that γ µν (2.35) is the effective metric on S 4 Λ .
3 Matrix model and fluctuations on fuzzy S 4 Λ Now we would like to make S 4 Λ dynamical, by considering as a background in the Yang-Mills matrix model Remember that we always work in the local frame at p, where any x µ can be dropped.
with a mass term as a regulator, and studying the fluctuation modes on S 4 Λ . We will later focus on the IKKT model [1] with D = 10. The classical equations of motion are We use the letter Y to indicate generic configurations, while X will indicate the fuzzy S 4 Λ background. Although the latter is not quite a solution of these equations, it was shown in [36] that quantum corrections (at one loop) can stabilize the radius of S 4 Λ for small positive µ 2 . A more refined one-loop analysis will be presented in section 6. Now consider fluctuations around some fixed background X a , where A a ∈ End(H) will be the dynamical degrees of freedom. Expanding the action expanded up to second oder in the A a , one obtains dropping the linear terms (for stable backgrounds). Hence the quadratic fluctuations A a are governed by the quadratic form where the f 2 term was canceled by adding a suitable Faddeev-Popov gauge-fixing term (choosing the Feynman gauge [37]) for the gauge fixing function Hence the fluctuations are governed by the "vector" (matrix) Laplacian is the SO(5) generator in the vector representation. The fluctuations A a entail fluctuations of the "flux" For backgrounds given by basic noncommutative branes M, this leads to noncommutative gauge theory, or equivalently to a theory of geometric deformations leading to some emergent gravity on the brane which seems to be different from general relativity [8,38]. However on the covariant S 4 Λ backgrounds, we will argue that (at least linearized) general relativity arises indeed from certain deformation modes, extended by a higher spin sector.

Decomposition into fluctuation modes
Global SO(5) notation. Given some deformation X a + A a of the S 4 Λ background, we want to identify the various fluctuation modes of the 5 fields A a . We can organize the tangential and radial fluctuations as follows, working mostly in the semi-classical limit Here and the functions ξ a , A b , A abc , κ ∈ C N (S 4 ) ⊂ End(H) play the role of tensor fields. The expansion in M correspond to expanding End(H) = ⊕(n, 2m) in terms of m. The ξ a and the A a are clearly tangential, and κ describes the radial fluctuations. We will only keep tensors of rank up to 3. The A a contribution is reminiscent of the standard parametrization in noncommutative gauge theory, and could be interpreted as u(1) × so(5)-valued gauge field (or more generally as U (so (5))-valued gauge field). Since the X a and M ab are tensor operators, there is an SO(5) action on these fields via etc., which leaves the background sphere invariant and implements the isometries on the tensor modes. In this sense, the theory to be elaborated will be "covariant". The extension to local gauge symmetries will be discussed shortly.

Now observe that the trace sector of A bcd
using (2.33), which is redundant with the ξ a modes. Therefore we should either impose that A bcd is traceless, or drop the ξ a modes (and the κ modes for the generic spheres S 4 Λ ). We will mostly choose the latter option.
Local SO(4) notation. To make the physical content more transparent, we will organize these fields further into 4D fields near some reference point p ∈ S 4 ("the north pole"). We will use greek indices µ, ν ∈ {1, ..., 4} for tangential components transforming as vectors under the local SO(4) around p ∈ S 4 , and latin indices a, b, ... ∈ {1, ..., 5} for the SO(5)covariant components. In particular, the fields will be locally expanded in powers of the SO(4)-covariant generators as in section 2.3, Θ µν = r 2 M µν and P µ : This organization gives the following modes up to the order under consideration here: where both A νρσ and A νρ = A νρ5 arise form the 5D fields A abc . We separate A µν into symmetric and antisymmetric (AS) parts, As discussed above, we can absorb ξ µ iñ Here P SD is the projector on the SD antisymmetric component. Similarly, the A 5 modes can absorb the radial κ modes for the generic spheres, but should be dropped for S 4 N since the corresponding fluctuations A µ ∼ P µ A 5 vanish.
We will see that the A µ describes a U (1) gauge field and h µν determines the metric fluctuations, while a µν does not seem to play a significant role. A µνρ is part of the gravitational sector. As discussed in the previous section, the h µν modes only exist on the generalized spheres S 4 Λ , while they vanish on the basic S 4 N due to the tangential constraint (2.36). It is important to keep in mind that (apart from the ξ µ and the κ deformations) these deformation modes are "internal" degrees of freedom, whose averages [.] 0 over the local fiber vanishes. Some of these deformations are sketched in figure 1. The only modes which change the embedding of S 4 in target space are the radial modes κ. The organization (3.17) is quite general and applies also to other covariant quantum spaces, even with Lorentzian signature. The full expansion into higher spin modes is obtained by allowing the A µ , ξ µ and κ fields to take values in the universal enveloping algebra of so (5) or Iso(4).

Gauge transformation
Consider gauge transformations with some gauge parameter Λ ∈ End(H). For fluctuations on a background Y a = X a + A a , this leads to the inhomogeneous transformation We can expand the gauge parameter in SO(5) generators as where Λ 0 , Λ ab ∈ C N (S 4 ). Clearly Λ ab M ab generates an x-dependent SO(5) transformation, and It is interesting to work out the explicit form of these transformations in the local 4D parametrization (3.17). We expand and define the individual transformations In the semi-classical limit, we can replace the commutators by Poisson brackets, and where φ = φ(X) indicates some scalar field, and θ µν is the undeformed Poisson tensor. Here we recalled that x µ = 0 = δ Λ x µ at p. Restricted to the lowest degree in θ, the δ v clearly acts as a diffeomorphisms on scalar fields φ(x), and δ Λ leads to local SO(4) rotations of tensors (which vanishes for scalar functions at the north pole). Applying this to the background X + A, we can read off the transformations of the tangential and radial perturbations Here we denote the local rotation of A µ by Λ ∈ so (4) with which extends to all the tensor legs of A µ in the expansion (3.17). Dropping contributions to the higher ξ µ modes which we don't keep track of, we obtain the following linearized gauge transformations for the 4D fields Separating A µ into the tensor components, this gives These can be understood as local SO(4) rotations generated by Λ µν (x), U (1) gauge transformations generated by Λ 0 (x), and infinitesimal diffeomorphisms generated by −v ρ ∂ ρ . The A µρσ transforms like a SO(4) gauge field. The inhomogeneous transformation of h µν under diffeomorphisms can be understood by anticipating that it plays the role of a linearized metric fluctuation g µν − h µν ; its transformation by v then gives which is the transformation of the metric tensor g µν − h µν under an infinitesimal diffeomor- Higher-order gauge transformations. The gauge transformations considered in (3.24) are only the lowest in a whole tower. Consider e.g. the transformations generated by This allows to gauge away the symmetric ξ µν modes. In contrast, one cannot gauge away the h µν modes. From a geometric point of view, the pure gauge modes correspond to Hamiltonian vector fields on CP 3 , and a systematic analysis is postponed for future work.
Gauge fixing. The gauge fixing was achieved by adding the Faddeev-Popov (or BRST) gauge fixing term −f 2 to the action, such that the explicit f 2 term in (3.4) is canceled. This ensures that the propagator is well-defined. The corresponding gauge fixing condition 0 = i[X a , A a ] is accordingly not a "hard constraint", but simply selects the physical Hilbert space or configuration space without redundancies.
Now consider the gauge fixing condition The radial contribution from κ is thus the gauge fixing condition is Separating the components, this leads to For a µν = 0, these reduce to the Lorentz gauge condition for A µ and A ρµν while the second condition reduces to ∂ µ h µν = 2 R ∂ ν κ.

Geometry: metric and vielbein
Undeformed background. Consider some scalar field φ = φ(X). The adjoint action of the basic matrices [X a , .] defines a derivative operator on φ, plays the role of a vielbein or frame. Using (2.40), we see that the tangential vielbeins play the dominant role, while the transversal component e 5 ∼ θ 5µ ∂ µ only contributes a small multiplicative factor Recalling the discussion in section 2.3, this vielbein arises from the bundle of (selfdual) 2-tensors 9 θ µν , which transform in the (1, 0) under SO(4) along the internal fiber S 2 . Hence e αµ is not a fixed frame on S 4 , but it is averaged out over the fiber, [e eν ] 0 = 0. We can now rewrite the formula for the metric (2.35) on S 4 Λ in a more suggestive way as follows This defines a fixed, well-defined metric on S 4 which is constant along the fiber S 2 , This is the key property which allows to reconcile covariance with noncommutativity. For generic S 4 Λ with large L R , theγ µν is replaced by the 5-dimensionalγ ab as in (2.37). Now consider general fields φ ∈ End(H), decomposed into a tower of higher spin (tensor) fields on S 4 as in (2.26). The adjoint action [X a , .] still defines a derivative operator φ, which however contains non-derivative terms which arise from commutators of the X α with the P µ generators in the expansion of φ. E.g. for φ = φ + φ µ P µ + φ µν M µν , we have This phenomenon will play a crucial role below. Nevertheless, the metric in the kinetic term for arbitrary fields is always obtained from the leading derivative contributions Deformed background. Now we include the fluctuations Y a = X a + A a . Since the kinetic term for (bosonic) fields always arise from contracted commutators we can read off the effective metric in the perturbed matrix model (up to a possible conformal factor, see below) Note that θ µν is no longer self-dual for the generalized spheres S 4 N . Then the treatment in this and the following sections should be generalized accordingly. However this will not lead to significant changes, and we stick to the self-dual case here for simplicity. and similarly for the 5D case with γ ab . This can be expressed in terms of an (over-complete) frame e a [A] = e aµ [A]∂ µ = D a (4.10) cf. [8,39]. Again the tangential contributions e α [A], α = 1, ..., 4 will provide the leading contribution. Recall the explicit form of tangential fluctuations Observe first where the non-derivative A µν term arises from the explicit P modes in A µ , similar as in (4.6). Using these expressions and dropping as usual [M αβ , φ] = O( x R ∂φ) in the local frame, we obtain 10 where the tangential vielbein e α [A] = e αν [A]∂ ν is obtained as (4.14) The transversal vielbein does not contribute to the linearized metric perturbations and can be dropped. Hence the tangential e α , α = 1, ..., 4 play the role of the effective vielbein. Using these results, the metric on a deformed S 4 Λ background including linearized perturbations is γ µν = {Y α , X µ }{Y α , X ν } = g αβ e αµ e βν =γ µν + δγ µν (4.16) where the linearized metric fluctuations are given by using (2.35), always raising and lowering indices with g µν . Note that the anti-symmetric contributions a µν drop out. After averaging over the fiber S 2 using (C.1), the contribution from the U (1) gauge field A ν also drops out since [θ βν ] 0 = 0, and we obtain The contribution from the rank 3 tensorÃ µρν is using ∆ 4 4 = r 2 R 2 as well as the self-duality ofÃ µρσ in the last indices. Note that k µν transforms as under local SO(4) gauge transformations, and the gauge condition (3.38) forÃ µρν implies

Thick spheres, extra dimensions and dimensional reduction
Since the underlying space O[Λ] is higher-dimensional, there are excitation modes in extra dimensions. Most of them give rise to higher spin modes, as discussed above. However for the generalized spheres S 4 Λ , there are also extra scalar modes, corresponding to the SO(5) Casimir R 2 . For long-range gravity, we will actually need very "thick" spheres S 4 Λ , with large n √ N (5.39). In order to obtain nevertheless 4-dimensional physics, we need to assume dimensional reduction, i.e. all wavefunctions are constant in these extra directions. More precisely, we assume that only the lowest non-trivial modes (3.17) in these directions are significant, essentially leading to gravity.
There are several possible justifications for this dimensional reduction. First, since (most of) these internal excitation modes lead to higher spin fields, their interactions will be suppressed by a large mass scale, which should arise from the scale of noncommutativity 11 . Furthermore, quantum effects will certainly induce some mass gaps, in particular for the radial direction, cf. [36]. Finally there is another mechanism to give masses to internal modes, by explicitly embedding extra dimensions in the matrix model, along the lines of [31,40,41]. This can easily lead to a large mass gap 12 in the fuzzy extra dimensions, and at the same time lead to an interesting low-energy gauge theory. For example, the 6 transversal matrices in the IKKT model could be identified with the generators of squashed CP 2 [41,42], since S 4 Λ is a CP 2 -bundle over S 4 N , see appendix A. This will be studied elsewhere. Here we will simply assume dimensional reduction to 4-dimensions, and elaborate the resulting 4-dimensional gravity. In any case, a significant asymmetry 0 n N of Λ = (n, 0, N ) is presumably essential to justify this dimensional reduction.

Effective metric and scalar fields
To properly identify the effective metric, consider scalar fields propagating on the deformed S 4 Λ background in more detail. The kinetic term for a (transversal) scalar field is To be specific, we use the Riemannian measure in target space. We can cast this into a covariant-looking form (cf. [8]) in terms of the effective metric which is a dimensionless Weyl rescaling of γ µν , and a field ϕ which has dimension mass via Here g YM is defined in (4.36), and the corresponding linearized metric fluctuation is obtained from (4.18) whereh µν = h µν + k µν . Then the Lorentz-gauge condition ∂ µh µν = 0 translates into the de Donder gauge for H µν , We consider H µν as a tensor field here, rising and lowering indices with g µν . Then the linearized coupling ofh µν to matter is given by is the energy-momentum tensor of ϕ, which satisfies ∂ µ T µν = 0.

Flux and field strength
Now consider the perturbation of the "flux" [Y µ , Y ν ] ∼ iΘ µν (Y ) given by θ µν := θ µν + θ µµ A µ ν g ν ν − θ νν A ν µ g µ µ = (δ µ µ + A µ ρ g µρ )θ µ ν (δ ν ν + A ν ρ g ρν ) (4.30) to linearized order. Since the A terms enter through one factor of θ µµ , they are naturally viewed as geometric deformation of the background θ µν →θ µν , which plays the role of the Poisson tensor in the deformed y µ coordinates. In contrast, the field strength enters via 2 factors of θ, and decomposes into the U (1) × Iso(4)-valued components Clearly F µν is the U (1) field strength of A µ , and R µν is the curvature of A µαβ M αβ viewed as so (4) connection. Furthermore, T µν can be related to the linearized spin connection.
Stack of branes and nonabelian gauge theory. As usual, SU (k) gauge fields can be obtained by considering a stack of coincident branes, expanding the fluctuations as A µ = A µ 1l + A µ,a λ a in terms of su(k) generators λ a . For the generic S 4 Λ spheres there is no need to do this by hand, since they can be interpreted as bundles over S 4 N with fibers being fuzzy coadjoint orbits of SU (3) (see Appendix A for more details, and [43] for an explicit example in a simplified setting). This means that some non-trivial gauge theory will arise automatically, whose structure is similar to the squashed brane configurations in [41,42], which in turn are quite close to the standard model. It is very remarkable that the S 4 Λ spheres seem to provide the right ingredients for both gravity and particle physics. For such nonabelian gauge fields arising on S 4 Λ , the u(k)-valued fluctuations should be expanded in terms of 13 A α =ẽ αν A ν (4.33) rather than A α = θ αν A ν . Hereẽ is the conformally rescaled dimensionless vielbein corresponding to the effective metric G µν (4.24). Then the correct coupling to the metric is recovered (cf. [8]), where the Yang-Mills coupling constant is defined as 14 Noting that the conformal factor drops out in the Yang-Mills action, the linearized coupling to the metric perturbation h µν gives is the energy momentum tensor of the gauge fields.

Classical action and equations of motion
In order to derive the equations of motion for the gravitational A µν , A µρσ and κ modes, we evaluate the semi-classical action up to quadratic order. The quadratic fluctuations are governed by the "vector" (matrix) Laplacian (3.6) Now consider the ansatz 15 (3.17) dropping the U (1) component for now. We can evaluate D 2 A in the semi-classical limit using the basic rules which are valid in the local ON frame at p. After some algebra (see appendix E) and using the antisymmetry of A νσρ in the last two indices, we obtain with (3.17) using ∆ 4 4 = r 2 R 2 . All other mixed terms such as [P σ θ µν g µµ θ µ ν M σ ρ ] 0 vanish. Thus provided either A νσρ or A νσρ is SD in the last 2 indices. Therefore the semi-classical action (3.4) quadratic in the A µν ,Ã νσρ and κ fields is To accommodate one-loop effects (see section 6), we introduced a factor α in the mixed term, which is α = 1 (5.10) for the present semi-classical action. Given in addition a coupling to matter of the form 16 (cf. (4.37), (4.28)) we obtain the equations of motion Here P 0 SD = 1 4 (δδ − δδ + ) is the projector on the SD antisymmetric component 17 , P 0 is the projector on the trace contributions ofÃ µρσ , and P hor is the operator exchanging horizontal indices in the mixed hook diagram (corresponding toÃ µρσ ) defined in Appendix D. This has eigenvalue P hor = −1 on the totally antisymmetric diagrams, and P hor = 1 2 on the mixed (hook) Young diagrams. In any case, the contribution of P hor and P 0 in the kinetic operator is negligible compared with , and we neglect it for simplicity, and we replace the equation forÃ νσρ by This implies Now assume that the antisymmetric component of A µν vanishes, a µν = 0. Then the gaugefixing condition (3.38) implies ∂ σ A νσ = 1 R ∂ ν κ, so that ∂ σÃ νσρ is symmetric in ν, ρ. Then the eom (5.12) for A µν is indeed consistent with for symmetric and conserved T µν . Putting all this together and recalling ∼ − ∆ 4 4 g , we obtain the following equations for the metric contributions k µν (4.19) and h µν and κ Here we define the mass which is just the background curvature unless µ 2 is significant. We expect from (5.16) that g T , so that we can neglect the κ contribution (except possibly for the longest "cosmological" scales). Then combining the first two equations gives It is now useful to separate k µν into a "local" and a propagating "gravitational" part, Then (5.18) reduces to To ensure stability, we should require m 2 k ≥ 0, which means Moreover we want to assume that m 2 is very small but positive 18 . In view of (2.38), this requires a generalized 4-sphere S 4 Λ with large n. Then the term m 2 h µν can be dropped, and we can solve (5.20) as Inserting this in the equation for h µν gives As long as the masses m 2 and m 2 k can be neglected, the first term is governed by the usual long-range 1 r 2 propagator (or rather Greens function), while the second term describes a sub-leading 1 r 4 correction 19 . Now we observe that the local contribution k (loc) µν ∼ T µν (5.19) can be dropped; this is merely a negligible local "contact" contribution to the metric, and has nothing to do with any long-distance gravitational effect. For example, to compute the gravitational effect of the sun at the location of the earth, we certainly have k (loc) µν = 0. Even for an observer located in a cloud of gas with some energy-momentum density T µν , the "local" contribution k (loc) µν = O(g 2 YM ∆ 4 T µν ) would still be negligible except for extremely high energy densities. Thus we will replaceh µν by the effective gravitational metric (5.24) Its physical significance depends on the distance scale: Regime G: Consider first the gravitational field for distance scales D 2 . This is the "gravity" regime, where the propagators can be replaced by massless ones. Then the eom for h µν and k (grav) µν reduce to

25)
18 As shown in [36], such a positive m 2 can in fact stabilize S 4 N . See section 6 for more details. 19 Observe that the second term is of order m 2 and the effective gravitational metric satisfies We will see that this amounts to the linearized Einstein equations. Note that both equations are compatible with the Lorentz gauge condition ∂ µ h µν = 0 = ∂ µ k µν . For α → 0, this regime always applies even for L R ≈ 0. T µν . This is the "cosmological" regime. Hence gravity ceases to operate for structures much larger than m 2 , as always in massive theories.
Intermediate regime. Finally, there may be an interesting intermediate regime provided Then h µν is exponentially damped, while k (grav) µν is still in its massless, propagating regime. Then we still get the Einstein equations but the effective Newton constant is reduced compared with (5.26). Formally, (5.20) admits the possibility that m 2 m 2 k ≈ 0 even for small L 2 R . Then this intermediate regime would provide long-range gravity even for an almost-basic S 4 Λ with n ≈ 0. However this would be fine-tuning, and it seems unlikely that this is compatible with the radial stabilization, cf. [36] and section 6.
Self-dual action. We conclude this section with the following interesting observation. Suppose the matrix model has an additional term such that it reduces to a selfdual Yang-Mills action 20 is averaged over the local fiber, and cf. (4.30). The first term is equivalent (modulo gauge fixing) to the quadratic action (5.9). The second term F µν F ρσ µνρσ as usual will not affect the local field equations, except for the mixed term between A µν and F µν , which after some algebra is using selfduality of A ναβ , its symmetry property and det θ = ( ∆ 4 4 ) 2 . This has indeed the same form as the mixed term in (5.9) (which is the reason for introducing the factor α in (5.9)). Similarly, using partial integration and the gauge fixing condition. The contribution from h ρα agrees up to a factor (-2) with (5.31). Thus the mixed term cancels in the selfdual Yang-Mills action, which reduces to (5.9) with α = 0. Then the regime G always applies, without the need for large L R . Together with the following section this implies the linearized Einstein equations always arise, without IR modification. We leave this as an interesting observation.

Curvature and linearized Einstein equations
Now we consider the curvature of the linearized effective metric viewed as a perturbation of the flat metric δ µν | p = g µν | p near p; recall that H µν was defined in (4.26) as trace-reverse ofh µν = h µν + k µν , or rather byh (grav) µν (5.24) as discussed above. Furthermore, assume that we are in the scaling regime G, where the masses m 2 and m 2 k can be neglected. Then consider the linearized Ricci tensor [44] Since H µν satisfies the de Donder gauge (4.27), it simplifies as with the Newton constant given by using (2.38). This entails the interesting reciprocity relation 21 Since the gravity regime G applies only for distances smaller than 1 m ≤ L R /α (5.21), we should require L pl L R /α. Therefore one of the following conditions must hold Thus we need either a self-dual action, or a very "thick" fuzzy sphere S 4 Λ . Notice that such a macroscopic "thickness" L R of S 4 Λ does not necessarily mean that space is effectively 5dimensional. This point was discussed in section 4.1, although a more detailed investigation is required to settle this. There is no such issue with dimensional reduction for the self-dual action (5.28), where c n can be very small. Now consider briefly the gravitational coupling of fermions. We only observe here that the matrix model Dirac operator can be rewritten in terms of "comoving" Clifford generatorsγ µ = Γ αẽ αµ (up to possibly a conformal factor), which encode the effective metric {γ µ ,γ ν } = 2G µν [8]. Together with supersymmetry [1], we expect that fermions couple properly to gravity in the present framework, however a detailed analysis is left for future work.
The above results show that the 4-dimensional (Euclidean, linearized) Einstein equations can emerge from the classical dynamics of fluctuations on fuzzy 4-spheres S 4 Λ in the Yang-Mills matrix model (3.1), provided certain conditions for Λ are met (i.e. large n), and dimensional reduction is justified. No explicit Einstein-Hilbert term is required. There are several contributions to the metric fluctuationh µν , so that the physics is richer than in general relativity. Most notably there is a long-distance cutoff of gravity set by some IR mass scale m −1 . The requirement of large n might be avoided for the self-dual action.
In any case, the long-wavelength modifications of gravity discussed above could be very interesting for cosmology, and it is tempting to relate this to some of the effects attributed to dark matter or dark energy. There will also be some additional gravitational modes arising e.g. from radial deformations κ. However, a more detailed analysis is required before these issues can be addressed.
One final but crucial question is whether the present mechanism survives quantization. The (preliminary and partial) analysis in the following section supports the conjecture that the quantization is well-behaved and preserves the above picture, for the IKKT model.

One-loop corrections from string states
As a first step towards a full quantum theory, we would like to study the one-loop effective action for the above gravitational fluctuations around a fuzzy S 4 Λ background. This should be done in the maximally supersymmetric IIB or IKKT model, which is the only model where the non-local UV/IR mixing in noncommutative field theory is mild (leading to 10dimensional IIB supergravity in target space). Until recently, such a 1-loop computation in terms of a mode expansion would be hopeless; already the one-loop effective action for the constant radius is quite involved [36]. However the integration method using string states [25] makes this task feasible. As a check of these methods we will first reproduce the results in [36], and then proceed to extract the leading 1-loop contributions to the effective action. For simplicity we will restrict ourselves to the basic fuzzy sphere S 4 N here.

The 1-loop effective potential for the IKKT model
We start with the bare bosonic action (3.1) for the background X supplemented by a mass µ 2 , to regularize possible IR singularities. Adding the fermions in the IKKT model, the one-loop effective action is defined by with a, b = 1, ..., 10, where 5) and the 2 log term arises from the ghost contribution. Note that the coupling constant g drops out from Γ 1loop due to supersymmetry. For µ = 0, the first non-vanishing term in this expansion is n = 4 due to maximal supersymmetry. However there are contributions of order Θ for µ 2 = 0 due to the soft SUSY breaking, which are important to stabilize the background.
The leading 4th order term is given by the following expression [37]: and the leading term in µ 2 is We will evaluate the trace over hermitian matrices in End(H) using the basic formula [25] Tr Here |y x| ∈ End(H) are string states 22 , which are built out of coherent states |x on M = CP 3 ≈ S 4 N × S 2 . The formula is exact for homogeneous spaces such as CP 3 . It follows by noting that the rhs of (6.8) is invariant under SO(5) L × SO(5) R , and so is the trace functional on End(H Λ ). The normalization of the measure 23 in the integrals cancels out, and we will choose it as product measure on S 2 × S 4 with unit volume of S 2 and the measure on S 4 is induced by the target space metric. For deformed M, (6.8) is expected to be an excellent approximation, as long as O is well localized.
The string states are very useful for loop computations, because they have approximate localization properties in both position and momentum; see [25] for a detailed discussion. In particular, We can therefore approximately evaluate the 1-loop integral as follows (suppressing the target space metric g ab ). First, we note that Γ 1loop;4 [X] vanishes identically for constant fluxes Θ = const. This is a reflection of the maximal supersymmetry of such backgrounds. Due to the SUSY cancellations, the interaction decays like r −8 , and it is bounded at short distances by the NC cutoff ∆ 2 . This means that the one-loop induced action is a weak short-distance effect on branes with dimension less than 10 (which is essentially IIB supergravity). We will compute its effect on the fluctuations on the fuzzy S 4 background below.
6.2 Vacuum energy of S 4 N Mass contribution. We start with the contribution of µ 2 (6.10): using (2.18). Note that S 4 denotes the unit sphere in this computation. Using dim H = 1 6 (N + 1)(N + 2)(N + 3), (6.15) we obtain This agrees precisely with the group-theoretical computation in [36]. This term describes the positive vacuum energy contribution due to the explicit SUSY breaking by the bosonic mass µ 2 , which scales like µ 2 r 2 .
Curvature contribution. Now we compute the µ 2 = 0 contribution We can fix x = (x, ξ) to be some fixed reference point on M ≈ S 4 × S 2 , where x µ are local coordinates on S 4 . We first compute the integration over η ∈ S 2 with y = x, which is the projection defined in (2.25) [f (x, η)] S 2 := 1 4π at any given x ∈ S 4 . Recalling the identity γ bc = 1 4 ∆ 4 P bc T (x) where P T is the tangential projector on S 4 ⊂ R 5 and using (C.1), this gives using the notation θ ab x = θ ab (x, η) etc., and x · y = x b y b , and P ab T (x)P ab T (y) = 3 + (x · y) 2 P ab T (x)P bc T (y)P cd T (x)P da T (y) = 3 + (x · y) 2 ε abcde x e ε abcdf y f = 24(x · y) . (6.20) We can now evaluate (6.17) as follows Again the last line agrees with the (more involved) group-theoretical computation in [36], providing further support for the coherent state approach. The present computation is not only shorter, it also allows to see more clearly the origin of the attractive interaction: It arises from nearly-local interaction among the N degenerate sheets at points x ≈ y ∈ S 4 . At coincident locations x = y, the cancellation is exact, because θ µν (x, ξ) is selfdual. In other words, the interaction is a residual attractive IIB supergravity effect which arises due to the curvature of S 4 . This also confirms the stabilization mechanism put forward in [36].

Fluctuations
Having gained confidence in the coherent state approach to 1-loop integrals on fuzzy S 4 , we turn to the 1-loop effective action for the fluctuation fields. This is of course a major task, and we will only consider the leading corrections to the kinetic terms for the lowest spin excitations of interest here.

Transversal fluctuations
Consider first the contributions from the transversal flux components F µa ∼ −i[X µ , A a ] where a = 5, ..., 10. It is easy to see that only the transversal fluctuations φ i in contribute to F µa . The general formula (6.12) for S 4 [δθ + δF] shows that the dominant interactions arise for points x = (x, ξ) and y = (x, η) on M = CP 3 with the same x ∈ S 4 . Then transversal fluctuations can arise only at quadratic (or higher) order, contracted as Here we assume that φ a = φ a (x) is constant along the S 2 fiber, so that F µa = θ µα ∂ α φ a (4.6), andT However, we claim that this quadratic contribution in φ cancels due to the averaging over S 2 , and the only non-vanishing contributions are higher-order interactions or higherderivative terms. To see this, note that the quadratic contribution would arise from Averaging over (ξ, η) ∈ S 2 × S 2 and using invariance under SU (2) L and therefore under SO (4) Therefore transversal deformations of the background do not acquire quadratic quantum corrections at one loop, up to possible subleading higher-derivative terms. As a check, S 4 [φ] vanishes for radial deformations A a = X a , whereT µν ∼ g µν . This is in contrast to tangential deformations, as we will see.

Tangential fluctuations
The one-loop effective action is given by The propagators act like a short-range delta function with normalization using the Riemannian measure, and M 4 indicates S 4 with radius R. Therefore the dominant contribution will come from local interactions with x = (x, ξ) and y = (x, ξ) denoting the same x ∈ M 4 but different points on the internal S 2 . We can therefore replace the integral over M 4 × M 4 by a single integral as follows For the tangential fluctuations, S 4 can be written locally using (6.12) in terms of (antiselfdual) flux components as follows: using (C.5). Here we used the self-duality of the background θ − = 0, while θ + + F + ≈ θ + for small fluctuations. As above, x, y denote the same x ∈ M 4 but different points ξ, η on the internal S 2 . This should be integrated over S 2 × S 2 for each x ∈ M 4 . Since m 2 > 0 whenever ξ = η, only the ASD components F µν − contribute, with a negative sign. Hence fluctuations F(x) ∈ (n, 0) which are constant along S 2 drop out, but all the higher spin fluctuations such as F(x) ∈ (n, 2) will contribute.
The gravity modes of interest here give rise to F µν ∈ (n, 2). These can be written as for ξ a ∈ S 2 and F µν a a 3-vector. Then we can write This means that for such F ∈ (n, 2), the 1-loop effective action at O(F 2 ) can be written as As shown in section 5.1, this can be absorbed in a renormalized action (5.9) for a suitable value of α = 1.
Clearly the maximal supersymmetry of the model protects the flat limit R → ∞ from large quantum corrections (i.e. from the non-local UV/R mixing), leading only to the above mild term. Note that there is no "cosmological constant" induced at one loop; in fact the very concept does not apply in this framework, which is based on matrix degrees of freedom rather than a fundamental metric. Only the background curvature (which we dropped) might lead to modifications in the linearized Einstein equation (5.36) which look like a cosmological constant. Hence the "cosmological constant" problem is replaced here by the question of stability of a background with sufficiently large R and small extra dimensions. These are hopefully feasible problems which need to be addressed in future work.

Conclusion and outlook
We have shown that the 4-dimensional (Euclidean, linearized) Einstein equations emerge from the dynamics of fluctuations on fuzzy 4-spheres S 4 Λ in Yang-Mills matrix models, in a certain regime and provided certain conditions are met. The resulting physics is richer than in general relativity, since there are several contributions to the metric. Most importantly, gravity is cut off at some long-distance scale m −1 or L R /α. Moreover, a tower of higher-spin fields arises on top of the gravitational modes, leading to a higher-spin theory. The present analysis is expected to capture the leading gravitational effects, since fields with spin larger than 2 should decouple at low energies. Thus the gravitational physics of the present model could be sufficiently close to general relativity at least for solar-system scales.
The conditions to obtain an interesting gravity are as follows: 1) the background must be a generalized "thick" fuzzy sphere S 4 Λ with n √ N , leading to a large scale L R /α which acts as an IR cutoff for gravity, and 2) dimensional reduction to 4 dimensions is justified. We discussed possible mechanisms for the latter. One obvious mechanism involves the radial potential which stabilizes S 4 N . Another possibility is to give VEV's to the transversal scalar fields along the lines of [41,47], leading to fuzzy extra dimensions. This is natural given the structure of S 4 Λ as bundle over S 4 N , and it would also provide an interesting symmetry breaking structure, leading to a low-energy gauge theory in the right ball-park of particle physics [42]. Yet another possibility is to have a self-dual Yang-Mills action 24 (5.28); then α = 0, and the extra dimensions (i.e. c n ) may be small (but non-zero; the basic fuzzy sphere S 4 N does not suffice). Anyway, it is intriguing that the generalization to S 4 Λ seems to provides the required ingredients for both gravity and interesting particle physics.
To clarify these conditions requires a more detailed treatment of the generic fuzzy spheres S 4 Λ (cf. appendix A), as well as an understanding of the effective potential for the extra dimensions which would arise at one loop. Assuming that these conditions can be met, the long-wavelength modification of gravity discussed above could be very interesting, as they might lead to behavior usually attributed to dark matter or dark energy. There will also be new effects due to additional modes arising e.g. from radial deformations κ.
Apart from the above conditions, there are other issues which need to be addressed before physical implications can be extracted. One is to find a suitable Minkowski version of the background. While most of the analysis will generalize, the proper choice of a covariant Minkowskian matrix geometry is not clear, and there are non-trivial issues related to the non-compactness of the Lorentz group 25 . Natural candidates would be based on a noncompact version of SO(6) (cf. [48]), or possibly some fuzzy de Sitter space [49,50].
The restriction to linearized gravity in this paper is clearly not essential. The model is fully non-linear, and much of the derivation would go through for perturbations on a non-trivial background. We simply have to make the replacement (4.33) in the general mode expansion (3.16), and perturbations around a non-trivial γ µν could be studied along the same lines, leading presumably to the full Einstein equations on S 4 Λ . Hence there is no obstacle for describing strong gravity in this manner.
For the IKKT matrix model, the quantization should be well-behaved, and the present mechanism provides a promising basis for a quantum theory of gravity with low-energy physics close to GR. The maximal supersymmetry protects backgrounds with large radius, and leads to a stabilization [36]. Moreover the non-local UV/IR mixing is mild in this model, and reduces to 10-dimensional supergravity in the bulk [1,25,51]. We have started this endeavour by computing the leading one-loop corrections for the simplest fuzzy 4-sphere, which lead to modified parameters of the action including α.
The relation of the IKKT model with IIB string theory also suggests an interesting general message: compactification of target space may not be needed, so that the vast landscape of string compactifications may be avoided. While IIB supergravity arises in the bulk upon quantization, this has nothing to do with the present mechanism for gravity, which is purely classical. The present mechanism should therefore not be confused with mechanisms to localize bulk gravity to the brane such as [52]. If it is possible to obtain also a (near-) realistic low-energy particle physics in this framework (e.g. along the lines of [41,42,47]), it would provide an extremely simple and attractive approach to a quantum theory of fundamental interactions.
A The classical geometry of the 4-spheres S 4 Λ The fuzzy 4-spheres under consideration are quantizations of the (co)adjoint orbits O[Λ] = {g · H Λ · g −1 ; g ∈ SU (4)} → su(4) projected to R 5 via the projection Π (2.12), The coadjoint orbit is a homogeneous space O[Λ] ∼ = SU (4)/K where K is the stabilizer of Λ. Here we discuss the classical geometry of these spaces and their harmonics. This is best understood in terms of the spinorial representation of su(4) ∼ = so(6) on C 4 . Let γ a be 4 × 4 hermitian gamma matrices of SO(5) with {γ a , γ b } = 2g ab for a, b = 1, ..., 5. To be specific, we choose the Weyl basis where Then a 4-dimensional representation of so (6) can be defined by the following generators [53] Σ µν : where µ, ν = 1, ... 26 For better readability we do not impose tracelessness here. This does not lead to significant changes.
The stabilizer of Λ is K = SU (3) × U (1), and clearly O[Λ] ∼ = CP 3 . By inspection of (A.2) we find tr(H Λ γ µ ) = 0, µ = 1, ..., 4 This defines our reference point x (0) ∈ S 4 (the "north pole"). It is easy to see using SO (5) invariance and the explicit form of the generators (A.3) that x a x a ≡ 5 a=1 The second identity expresses self-duality. The stabilizer group of where SU (2) L acts on the +1 eigenspace of γ 5 . Hence there is a fiber of points x ∈ CP 3 over each point x ∈ S 4 , which at the reference point x (0) is obtained by acting with SU (2) L on |ψ 0 . These fibers are resolved by the functions m µν , µ, ν = 1, ..., 4, which define a tangential SD rank 2 tensor (or a 2-form) on S 4 with m µν m µν = 4R 2 N . (A.11) These define 2 independent functions, which describe the internal S 2 fiber of S 4 N ∼ = CP 3 over S 4 . However, the "momentum" functions p µ vanish for any point on the fiber over x. Hence there are no independent modes F µ (x)p µ on the basic sphere S 4 N . Another way to see this is via the Poisson bracket identity 0 = {x b x b , x a } = 2x b m ba (A.12) since x b x b = R 2 N for the basic fuzzy 4-sphere (but not for the generalized ones). At the north pole, this gives p µ = 0. Moreover, the following identity of so(6) tensors holds (which we refrain from computing here explicitly). Hence in contrast to the basic S 4 N , the "momentum" functions p µ are independent, so that the modes F µ (x)p µ are non-trivial. Similarly, the radial function R 2 = x a x a = 1 4 tr(Ξ ⊗ Ξ γ a ⊗ γ a ) = 1 4 tr Ξ ⊗ Ξ (−1l + 2P + 8P 1 ) = 1 4 − (trΞ) 2 + 2tr(Ξ 2 ) + 16N n( ψ 0ψ1 )( ψ 0 ψ 1 ) = 1 4 − (N + n) 2 + 2(N 2 + n 2 ) + 16N n|τ | 2 , τ = ψ 0 ψ 1 ∈ C = 1 4 N 2 − N n + n 2 + 16N n|τ | 2 , τ = ψ 0 ψ 1 ∈ C (A. 18) for Ξ ∈ O[Λ] using (B.1), where P is the permutation operator acting on C 4 ⊗ C 4 . Now the point is that τ is not invariant under SO (6), so that the spectrum of R 2 lies in an interval [R 2 min , R 2 max ] peaked around R 2 N = N 2 4 . This means that the generic 4-spheres S 4 Λ are "thick" spheres, with {R 2 , x a } = 0. This essential for the existence of independent momentum functions p µ , which are the basis of the present mechanism for gravity. Finally we note that the identity (A.13) still holds approximately, in the form 4 µ=1 m µa m µb = R 2 N P ab T + t ab (A. 19) where P ab T = δ ab − 1 x a x b is the tangential projector on S 4 ⊂ R 5 , and t ab = O n N arises from (0, 2, 0) modes in C ∞ (S 4 Λ ).

(C.1)
This also applies to S 4 N , and to S 4 Λ as long as N n i . Furthermore since θ µν is self-dual, we can write θ µν (ξ) = r 2 θ µν a J a (ξ) (C. 2) where J a are the generators of the internal fuzzy sphere S 2 N +1 , which in the semi-classical limit are functions J a : S 2 → R 3 on S 2 with radius given by θ µν θ µν = 4r 4 J a J a ∼ N 2 r 4 (C.3) using θ µν a θ b µν = 4δ b a .