The fuzzy 4-hyperboloid $H^4_n$ and higher-spin in Yang-Mills matrix models

We consider the $SO(4,1)$-covariant fuzzy hyperboloid $H^4_n$ as a solution of Yang-Mills matrix models, and study the resulting higher-spin gauge theory. The degrees of freedom can be identified with functions on classical $H^4$ taking values in a higher-spin algebra associated to $\mathfrak{so}(4,1)$, truncated at spin $n$. We develop a suitable calculus to classify the higher-spin modes, and show that the tangential modes are stable. The metric fluctuations encode one of the spin 2 modes, however they do not propagate in the classical matrix model. Gravity is argued to arise upon taking into account induced gravity terms. This formalism can be applied to the cosmological FLRW space-time solutions of [1], which arise as projections of $H^4_n$. We establish a one-to-one correspondence between the tangential fluctuations of these spaces.


Introduction
In the present paper we continue the exploration of 4-dimensional covariant fuzzy spaces and their associated higher-spin gauge theories, as started in [2,3]. These are non-commutative spaces which allow to reconcile a quantum structure of space(-time) with covariance under the maximal isometry. In contrast, quantized Poisson manifolds such as R 4 θ [4,5] are not fully covariant, as an explicit tensor θ µν breaks the covariance. In previous work [2,3], gauge theory on the fuzzy 4-sphere S 4 N was studied in detail, starting from the observation that S 4 N is a solution of Yang-Mills matrix models supplemented by a mass term, cf. [6]. Here we extend this analysis to fuzzy H 4 n , which is a non-compact quantum space preserving an SO(4, 1) isometry, also known as Euclidean AdS 4 . For other related work on covariant quantum spaces see e.g. [7][8][9][10][11][12][13][14]. The motivation for this work is two-fold: first, we want to develop a formalism to study gauge theory on H 4 n along the lines of usual calculus and field theory, in order to facilitate the interpretation of the resulting models. While S 4 N allows to use a clean but less intuitive organization of fields into polynomials corresponding to Young diagrams, the non-compact nature of H 4 requires to develop a calculus as well as field formalism reminiscent of the conventional treatment. We will achieve this goal, and obtain results analogous to the compact case but in a more transparent manner. The second motivation is to set the stage for a similar analysis of the cosmological fuzzy space-time solutions M 3,1 n found in [1,15]. These FLRW-type space-times have very interesting physical properties such as a regularized Big-Bang-like initial singularity and a finite density of microstates. M 3,1 n can obtained from the present H 4 n via a projection, which not only leads to a Minkowski signature, but also reduces the symmetry to SO (3,1). Since the group theory becomes weaker, it seems advisable to consider first the simpler (Euclidean) case of fuzzy H 4 n . We establish the relevant formalism in this paper, and moreover provide some explicit links between the modes on H 4 n and M 3,1 n . One of the most interesting features of 4-dimensional covariant fuzzy spaces is the natural appearance of higher spin theories. This can be understood by recalling that these spaces are quantized equivariant S 2 -bundles over the base space (i.e. S 4 or H 4 here), where the fiber is given by the variety of self-dual 2-forms on the base. The equivariant structure implies that would-be Kaluza-Klein modes transmute into higher-spin modes. Taken as background solution in matrix models, such as the IKKT model, one obtains a higher-spin gauge theory as effective theory around the 4-dimensional covariant fuzzy spaces. As a remark, the structure is reminiscent of twistor constructions, see also [16]. Let us describe the results of this paper in some detail. Starting from the classical as well as fuzzy geometry of the hyperboloid H 4 , we develop a calculus, solely based on the Poisson structure, to organize the fuzzy algebra of functions on H 4 n into SO(4, 1) irreducible components. We further establish a map between the modes in the irreducible components, suggestively called spin s fields, and conventional (rank s) tensor fields on H 4 . Having understood the "functions" on H 4 n , we proceed by considering H 4 n as background in the IKKT matrix model. As a first result, we classify all (tangential) fluctuation modes at a given spin level and exhibit their algebraic features. Subsequently, we are able to diagonalize the kinetic term in the action governing the fluctuations. Remarkably, the kinetic terms for all tangential fluctuations are non-negative such that no instabilities in the tangential sector exist.
Having in mind emergent gravity scenarios, we derive the associated graviton modes for spin 0, 1 and 2 fields. The spin 0 and spin 2 contributions satisfy the de Donder gauge, and at spin 2 one graviton mode emerges from the tangential sector. However, while the underlying modes do propagate, the graviton turns out to behave like an auxiliary field, and does not propagate at the classical level. The reason is that the field redefinition required for the graviton cancels the propagator, similar as in on S 4 N [2]. Nevertheless, our results are interesting and useful. First of all, since classical GR is not renormalizable, it should presumably be viewed as a low-energy effective theory. Then the starting point of an underlying quantum theory should be quite different from GR at the classical level, as in our approach, and gravity may be induced by quantum effects [17,18]. This is the idea of emergent gravity. The present model may well realize this idea, since the basic framework is non-perturbative and well suited for quantization (in particular the maximally supersymmetric IKKT model), and the required spin 2 fluctuations do arise naturally. The extra degrees of freedom may or may not help, but certainly covariance provides a significant advantage compared to other related frameworks, cf. [19]. In particular, it is remarkable that no negative or ghost-like modes appear in the tangential modes. Perhaps the most interesting perspective is the extension to the cosmological space-times M 3,1 . We will establish a one-to-one correspondence of the tangential modes on H 4 n to the full set of fluctuations on M 3,1 . Since the tangential modes on H 4 n are stable and free of pathologies (in contrast to off-shell GR), it seems likely that the Minkowski setting on M 3,1 provides a good model, too. In fact, the presence of negative radial modes on H 4 n would require to implement a constraint in the matrix model, which may spoil supersymmetry. This is not needed for M 3,1 , which provides further motivation for including a discussion of M 3,1 here. However, to keep the paper within bounds, we postpone the details for this case to future work. The paper is organized as follows: We start with a discussion of the classical geometry underlying H 4 n in section 2, before discussing fuzzy H 4 n in detail in section 3. In particular, we introduce a calculus suitable for decomposing the algebra of functions into modules of equal spin. The details of the decomposition and the properties of the irreducible modes are provided in section 4. Having established the fundamentals of fuzzy H 4 n , we explore the fluctuations around an H 4 n background in the IKKT matrix model in section 5. We pay particular attention to the classification of tangential fluctuations, and explicitly diagonalize their kinetic term. Subsequently, the gravition modes are identified and their equation of motions are derived. Before concluding we briefly explore the projection of H 4 n to the Minkowskian M 3,1 n in section 6. Finally, section 7 concludes and provides an outlook for future work. Relevant notation and conventions as well as auxiliary identities and derivations are collected in appendices A-D.
2 Classical geometry underlying H 4 n The classical geometry underlying fuzzy H 4 n is CP 1,2 , which is an S 2 -bundle over the 4hyperboloid H 4 . More precisely, CP 1,2 is an SO(4, 1)-equivariant bundle over H 4 as well as a coadjoint orbit of SO (4,2). This means that the local stabilizer group SO(4) acts nontrivially on the fiber S 2 , leading to higher-spin fields on H 4 , and a canonical quantization exists. The construction is similar to twistor constructions for Minkowski space.
Furthermore, the following SO(4, 2) identities hold: where J i describes the internal S 2 . This exhibits the structure of CP 1,2 is an SO(4, 1)equivariant bundle over H 4 . The fiber S 2 is generated by the local SU (2) L , while SU (2) R acts trivially.

SO(3, 1)-invariant projections and Minkowski signature
So far we have constructed H 4 and H 2,2 , but not a space with Minkowski signature yet. Space-times with Minkowski signature can be obtained by SO(3, 1)-covariant projections of the above hyperboloids. Explicitly, consider the projections figure 1. In section 6, the image M 3,1 ⊂ R 3,1 of Π x serves as cosmological FLRW space-time with k = −1, as discussed in [1]. In contrast, t µ is interpreted as internal space related to translations.
3 The fuzzy hyperboloid H 4 n Now we turn to the central object of this paper: the fuzzy hyperboloid H 4 n . H 4 n is a quantization of the bundle CP 1,2 over H 4 , which respects the SO(4, 2) structure and the projection to the base space H 4 . This is natural because CP 1,2 is a coadjoint orbit of SO(4, 2) via (2.11). As such CP 1,2 is equipped with a canonical SO(4, 2)-invariant Poisson (symplectic) structure; whereas on H 4 no such structure exists. H 4 n was first discussed in [21], and it serves as starting point for a quantized cosmological space-time in [15]. As for any coadjoint orbit, fuzzy H 4 n can be defined in terms of the operator algebra End(H n ), where H n is a suitable unitary irrep of SU (2, 2) ∼ = SO(4, 2). The representation is chosen such that the Lie algebra generators M ab ∈ End(H n ) generate a non-commutative algebra of functions, interpreted as quantized or fuzzy CP 1,2 n . The M ab are naturally viewed as quantized coordinate functions m ab (2.13) on CP 1,2 . Fuzzy H 4 n is then generated by Hermitian generators X a ∼ x a , which transform as vectors under SO(4, 1) ⊂ SO(4, 2), and are interpreted as quantized embedding functions (2.4). This will be made more explicit through an oscillator construction, which allows to derive all the required properties. To define fuzzy H 4 n explicitly, let η ab = diag(−1, 1, 1, 1, 1, −1) be the invariant metric of SO(4, 2), and let M ab be the Hermitian generators of SO(4, 2), which satisfy We choose a particular type of (discrete series) positive-energy unitary irreps 5 H n known as minireps or doubletons [22,23]. Remarkably, the H n remain irreducible 6 under SO(4, 1) ⊂ SO (4,2). Moreover, the minireps have positive discrete spectrum where the eigenspace with lowest eigenvalue of M 05 is an n + 1-dimensional irreducible representation of either SU (2) L or SU (2) R . Then the Hermitian generators [M ab , Because the restriction to SO(4, 1) ⊂ SO(4, 2) is irreducible, it follows that the X a live on a hyperboloid, with some R 2 to be determined below. Since X 0 = rM 05 > 0 has positive spectrum, this describes a one-sided hyperboloid in R 1,4 , denoted as H 4 n . Analogous to fuzzy S 4 N , the semiclassical geometry underlying H 4 n is CP 1,2 [21], which is an S 2 -bundle over H 4 carrying a canonical symplectic structure. In the fuzzy case, this fiber is a fuzzy 2-sphere S 2 n . We work again in the semi-classical limit. We also note the following commutation relations The negative sign arises from η = diag(−1, 1, 1, 1, 1, −1), and X is not positive definite. 5 Strictly speaking there are two versions H L n or H R n with opposite "chirality", but this distinction is irrelevant in the present paper and therefore dropped. 6 This follows from the minimal oscillator construction of H n , where all SO(4, 2) weight multiplicities are at most one, cf. [22,24,25].

Fuzzy H 2,2 n and momentum space
As in the classical case (2.27) and for later purpose, we also define where R r T 5 = −X 4 . As the restriction of H n to SO(3, 2) ⊂ SO(4, 2) is irreducible, the operators (3.7) satisfy the constraint cf. (2.28). This is the quantization of the hyperboloid H 2,2 ⊂ R 3,2 with intrinsic signature (+, +, −, −) of section 2.2 and becomes Lorentzian via the projection (2.3). The commutation relations are which justifies to consider T µ as translation generators, and Note the different signs in (3.10) and (3.6), which arise from of η 55 = −1 = −η 44 .

Oscillator realization, minireps and coherent states
The Hilbert space H n is a highest-weight unitary representation of SU (2, 2), which can be obtained by quantizing the spinorial construction of CP 2,1 in (2.1). For the quantization one replaces the classical 4-component spinor ψ α by 4 operators, which satisfy The associated bilinears realize the Lie algebra (3.1) of SO(4, 2), due to The M ab are self-adjoint operators, since As a consequence, they implement unitary representations of SU (2, 2) on the Fock space F = span{ψ . . .ψ|0 } of the bosonic oscillators, which decomposes into an infinite number of irreducible positive energy unitary representations H Λ . The oscillator algebra (3.16) can be realized explicitly as follows (cf. [23,26]): Consider bosonic creation and annihilation operators a i , b j which satisfy Using the a i , b j we form spinorial operators with Dirac conjugatesψ as required, and the SO(4, 2) generators are (3.24) The generators of SU (2) L and SU (2) R are defined by and the time-like generator X 0 (or the "conformal Hamiltonian" E) is given by is invariant. The non-compact generators are given by linear combinations of creation and annihilation operators of the form a † i b † j and a i b j .
Minireps. The simplest class of unitary representation has lowest weight space given by the Fock vacuum a i |0 = 0 = b i |0 , which defines [26] |Ω := |1, 0, 0 =: |0 , This gives the doubleton minireps built on the lowest weight vectors Acting with all operators of the form a † i b † j of L + on |Ω , one obtains positive energy discrete series UIR's H Λ of U (2, 2) with lowest weight Λ = E, n 2 , 0 and Λ = E, 0, n 2 . We will largely ignore the distinction and denote both as H n . These are known as minireps of so(4, 2), because they are free of multiplicities in weight space 7 . They correspond to fields living on the boundary of AdS 5 . The minireps remain irreducible under SO(4, 1) as well as SO (3,2), and they can be interpreted as massless fields on AdS 4 , or as conformal fields on Minkowski space. The lowest weight state E, 0, n 2 of H n generates a (n + 1)dimensional irreducible representation of either SU (2) L or SU (2) R with degenerate X 0 , naturally interpreted as fuzzy S 2 n . Comparing the above oscillator construction (3.17) with (2.13), it is manifest that for each H n , with n > 0, the M ab generators can be interpreted as quantized embedding functions This provides the quantization of the coadjoint orbits (2.11), which defines fuzzy CP 1,2 n . Since X 0 ≥ 1, they should be viewed as quantized bundles with base space H 4 n described by X a , and fiber S 2 n , for n = 1, 2, 3, . . .. The implicit constraints defining these varieties will be elaborated below. For n > 0, these spaces have been briefly discussed in [16,21], and we will mostly focus on that case. The minimal n = 0 case is different, but also very interesting, and we discuss it in some detail in appendix C.2.
Coherent states and quantization. The above discrete series irreps H n provide a natural definition of coherent states |m = g · |Ω ∈ H n , which are given by the SO(4, 2) orbit through the lowest weight state |Ω . The set of coherent states forms a U (1)-bundle over CP 1,2 , and allow to recover the semi-classical geometry of CP 2 as S 2 -bundle over H 4 via m ab = m| M ab |m . In particular, the lowest weight state is located at the reference point Ω| X a |Ω = x a ξ = (R, 0, 0, 0, 0), see (2.6). The local SO(4) generators M ij act on the coherent states over ξ in a spin n 2 irrep. These coherent states |m also provide a SO(4, 2)-equivariant quantization map from the classical space of functions on CP 1,2 to the fuzzy functions End(H n ): where |m is a coherent state 8 , and dµ is the SO(4, 2)-invariant measure. For polynomial functions, this corresponds to Weyl quantization, mapping irreducible polynomials P (m ab ) to the corresponding totally symmetrized polynomials P (M ab ); in particular Q(m ab ) = M ab . Likewise, square-integrable functions on CP 1,2 are mapped to Hilbert-Schmidt operators in End(H n ). We expect 9 that the map Q is surjective, and that all "reasonable" (e.g. square-integrable or Hilbert-Schmidt) harmonics in End(H n ) can be obtained as quantizations of higher-spin harmonics on H 4 via Q. This will be used below.

Algebraic properties of fuzzy H 4 n
Using the aforementioned oscillator realization, one can derive a number of useful identities for the above operators on H n ; we refer the reader to appendix C for the details. To begin with, consider the SO(4, 1)-invariant radius operator Since H Λ is irreducible under so(4, 1), it must follow that R 2 ∼ 1l. Indeed, one finds where n = |N + 2| = 0, 1, 2, . . .. Note that R 2 is positive for n = 0, 1, which seems strange because X 0 is positive. However, this is a quantum artifact, and the expectation values X a under coherent states still sweep out the usual H 4 . Additionally we compute the quadratic SO(4, 1) and SO(4, 2) Casimir operators We note that (3.35) agrees with [21]. Further identities can be obtained from the so(6) C identity (B.9), which entails This implies the so(4, 1) relation These correspond to (2.17), (2.18). Moreover, one finds To summarize, we have found counterparts for all relation of the classical geometry in section 2.1, which vindicates the choice of representation H n .

Wave-functions and spin Casimir
Given a representation H n of SO(4, 2), the most general "function" in End(H n ) can always be expanded as follows which transform in the adjoint representation M ab → [M ab , ·] of so(4, 2). The φ ab (X) will be interpreted as quantized tensor fields on H 4 , which transform under SO(4, 1). We define an SO(4, 2)-invariant inner product on C via For polynomials generated by the X a , this trace diverges. However this is only an IRdivergence, and we are mainly interested in normalizable fluctuations corresponding to physical scalar fields. Technically speaking, we will be working with Hilbert-Schmidt operators in End(H). These can be expanded into modes obtained by decomposing End(H) into unitary representations of the isometry group SO(4, 1) of the background. We will see that the expansion (3.42) is truncated at n generators M ab .
Spin Casimir. To proceed, we require a characterization of the above SO(4, 1) modes in terms of a Casimir operator which measures spin. One can achieve this by the SO(4, 1)invariant which measures the spin along the S 2 fiber. To understand this, we locally decompose so(4, 1) , for example at the reference point (2.6), into so(4) generators M µν and translation generators P µ = 1 R M µ0 . Then C 2 [so(4, 1)] = −R 2 P µ P µ + C 2 [so(4)], and R 2 P µ P µ ∼ −r −2 [X µ , [X µ , ·]] if acting on functions φ(x), cf. (3.69). Therefore S 2 ∼ C 2 [so(4)] should vanish on scalar functions φ(X) on H 4 , but not on higher-spin functions involving θ ab . We will see that this is indeed the case, and End(H n ) contains modes up to spin n as measured by S 2 (3.55). We also observe (3.45) Note that S 2 , , and C 2 [so(4, 2)] commute and can be diagonalized simultaneously.
Higher-spin modes on H 4 n . To determine the spectrum of S 2 for the modes in (3.42) we first prove the following identity for any f ∈ C: where {·, ·} + denote the anti-commutator. To see this, consider Similarly, and adding them yields (3.46). Next, starting from this identity immediately implies for totally symmetrized polynomials P n (X) in X. More generally, we show in appendix C.1 that this holds for any scalar field φ on H 4 quantized via coherent states, i.e.
As a next step, we consider the higher-spin fields. Using we find with quantized functions φ ab (X) on H 4 in (3.51), etc. We can similarly compute S 2 for any irreducible polynomial function in M ab , and obtain where P α ∼ is a 2-row rectangular Young projector. The restriction to these Young diagrams follows from the commutation relations (3.1) and the self-duality relation (3.40). This leads to the decomposition where C s is the eigenspace of S 2 = 2s(s + 1). We refer to appendix C.1 for the details. Of course the C s contain also forms of the type φ α (X) Ξ s α . However since the multiplication does not respect the grading, we can only say that C s is the quantization of tensor fields φ α (x) taking values in the vector space spanned by Ξ α , where Ξ α denotes both the polynomials in M ab and m ab . We remind the reader that Q (3.32) respects so(4, 2). The truncation 10 at n follows provided Q is surjective, since the corresponding classical expressions (3.56) with s > n are annihilated by Q. In fact they correspond to spin s > n irreps of the local SO(4), which are not supported by the local fiber spanned by the coherent states, which is a fuzzy 2-sphere S 2 n . See appendix C.1 for more details. This is a very remarkable structure, which leads to higher-spin fields on H 4 truncated at spin n. For small n, the uncertainty scale L 2 NC ≈ R 2 , see (3.59), is set by the curvature scale of H 4 ⊂ R 1,4 , so that the space is far from classical. Nevertheless, the case of small n may be interesting after projection to the cosmological space-time M 3,1 as discussed in section 6.
The n = 0 case. In that case, (3.40) gives which is a relation in the Joseph ideal [27]. Then the M ab , a, b = 0, . . . , 5 generate Vasiliev's higher-spin algebra associated to so(4, 2). However here we will not aim for a higher-spin theory on AdS 5 , but reduce H n for n = 0 to the so(4, 1) generators M ab , a, b = 0, . . . , 4, and the remaining X a generators. Then the M ab , a, b = 0, . . . , 4 satisfy relations which are locally similar to the hs algebra of so(4, 1), while the X a generate the underlying space.

Semi-classical limit and Poisson calculus
Now consider the semi-classical limit of fuzzy H 4 n , which is obtained for large n, and is indicated by ∼. Then X a ∼ x a and Θ ab ∼ θ ab , and the above relations on H 4 n reduce to x a x a = −R 2 , , (3.58a) where the scale of non-commutativity is Here is the Euclidean projector on H 4 (recall that H 4 is a Euclidean space). Hence the algebra of functions on fuzzy H 4 n reduces for large n to the algebra of functions on the classical Poisson manifold CP 1,2 , as described in section 2.1. We denote the eigenspaces of S 2 again with C s , which are now modules over the algebra C 0 = C(H 4 ) of functions on H 4 , thus encoding the structure of a bundle over H 4 . From now on we work in the semi-classical limit. The bundle structure can be made more explicit by writing as in (2.25), where η ab i are the tangential self-dual t'Hooft symbols; "tangential" follows from x a θ ab = 0. The J i transform as vectors of the local SU (2) L ⊂ SO(4), and describe the internal S 2 fiber.
Derivatives. It is useful to define the following derivations (cf. [2]) which are tangential x a ð a = 0, satisfy the Leibniz rule, and are SO(4, 1)-covariant. Equivalently, In particular, the following holds: as shown in appendix D. The first line shows that ð act as isometries on functions, such that the ð a can be viewed as a set of five Killing vector fields on H 4 with Lie bracket given by (3.65). Furthermore, We also note that the SO(4, 1) rotations of scalar functions are generated by {M ab , ·}, which can be written as To see this, it suffices to verify the action on the x c generators, since both sides are derivations. Finally, the semi-classical limit of the operator (3.6) can be expressed in terms of the derivatives as follows: for any φ ∈ C.
Connection. We define an SO(4, 1)-covariant connection on the module C (4.1) by [2] etc. if φ a , φ ab are tangential. Comparing with (3.66) and using (3.67) it follows that the connection is compatible with θ ab , i.e.
and ∇ a P bc = 0. The associated curvature is computed in appendix D. 15, and reduces to the Levi-Civita connection on tensor fields. Thus H 4 n is a quantum space which is fully SO(4, 1)-covariant, and we have found a calculus which is defined solely in terms of the Poisson bracket, i.e. the semi-classical limit of matrix commutators. This is very important for the present non-commutative framework.
Averaging over the fiber. There exist a canonical map defined by integrating over the fiber at each x ∈ H 4 . This projects the functions on the total space to functions on the base space. On fuzzy S 4 N , this averaging can be defined in terms of a SO(5)-invariant projection to some sub-space of End(H). For H 4 n , [·] 0 is nothing but the projection to S 2 = 0 i.e. to C 0 , as discussed below. Explicitly, the averaging [·] 0 over the internal S 2 is given by One can generalize the averaging to higher powers of θ ab , e.g. [2] [θ ab θ cd θ ef Alternatively, one could proceed to define a star product for functions on H 4 , which is presumably commutative, but not associative, in analogy to the case of S 4 N [11]. On the other hand, for n = 0 there is nothing to project, and the full algebra of functions on H 4 is non-commutative and associative without extra generators.
Integration. As for any quantized coadjoint orbit, the trace on End(H) corresponds to the integral over the underlying symplectic space, defined by the symplectic volume form. Explicitly, replacing the ill-defined fraction dim(H) Vol(H 4 ) with the symplectic volume form dµ, which reduces to ρ on H 4 . This is best seen via coherent states (3.32). We will often drop dµ and Q in the semi-classical limit. Finally, note that the ð a are not self-adjoint under the integral, but

Functions, tensors and higher-spin modes
We have seen that the algebra End(H n ) of fuzzy H 4 n reduces in the semi-classical limit to the algebra of functions on CP 1,2 . The results of section 3.5 provide a more detailed decomposition of C into modules (3.55) over the algebra of functions C 0 on H 4 , due to (3.46). This means that C is a bundle over H 4 , whose structure is determined by the constraints (2.17), (2.21) and (2.24). An explicit description is given by the one-to-one map 11 Here Γ (s) H 4 denotes the space of totally symmetric, traceless, divergence-free rank s tensor fields on H 4 , which are identified with (symmetric tangential divergence-free traceless) tensor fields φ (s) a 1 ...as with SO(4, 1) indices, as discussed in section 4.2 and in [2]. The inverse map of (4.2) (up to normalization) can be given by which is symmetric due to [·] 0 , as well as traceless, divergence-free and tangential. These statements are analogous to the results in [2].
Some comments on the map (4.2) are in order. We show in sections 4.0.1-4.0.3 that pure divergence modes would be mapped to zero by (4.2). Injectivity will be shown below by establishing (4.3). To see surjectivity, it suffices to consider the vicinity of a chosen reference point, for instance (2.6). Then polynomial functions suffice to approximate any element in C s . Then the so(4, 2) representation theory allows to characterize all polynomials in C s uniquely by Young diagrams, as explained in detail in [2, section 3]. These in turn are captured by the map (4.2), and an alternative inverse map can be used [2] C s φ (s) which is equivalent to (4.3) up to normalization. Hence C s encodes one and only one irreducible spin s field on H 4 , given by square-integrable tensor fields on H 4 . The generators Ξ β form a basis of irreducible totally symmetric polynomials in m ab , i.e. of Young tableaux As in [2], hs is closely related to the higher-spin algebra of Vasiliev theory 12 . Hence C can be viewed as functions on H 4 taking values in hs.

Spin 1 modes
The unique spin 1 field is encoded in φ ab m ab . According to the above statements, it can be expressed in terms of a tangential, divergence-free tensor field φ a ∈ C 0 on H 4 , i.e.
Given such a φ a , we define which encodes the field strength of the vector field. This is not tangential, but . The generalization of this formula for higher-spin is discussed below. If φ a is an irrep of SO(4, 1), we may abbreviate this as Pure gauge modes. Finally, one can verify that forφ a = ð a φ, the associated "field strength" tensor is F ab ∝ {θ ab , φ}, but the field strength form φ (1) vanishes identically: using (D.3). This expresses the gauge invariance (or irreducibility) of φ (1) .

Spin 2 modes
Similarly, spin 2 modes can be realized in terms of a tangential, divergence-free, traceless, We define the associated "potential form" which can be viewed as so(4, 1)-valued one-form with (4.14) Note that φ (2) c is indeed tangential, The so(4)-valued components of φ (2) a correspond to the spin connection, while its translational components reduce to φ ab , as on fuzzy S 4 N [2]. The "field strength form" corresponding to φ noting that the ð d θ cb terms drop out for for traceless, tangential φ ab , using (3.66). This encodes the linearized Riemann curvature tensor associated to φ ab , Although the ð e do not commute among another, their commutator is radial due to (3.67), i.e.
Hence the tangential components of P R ae;bc [φ] coincide with the usual linearized Riemann tensor. The connection form φ generalizing (4.9). Here, we defined Similarly to the spin 1 case, (4.20) could be obtained via formula (3.76); however, we provide a more transparent derivation by means of an inner product below. If the underlying tensor φ ab is an irrep of SO(4, 1), we may abbreviate this as Spin 2 pure gauge modes. Again, consider a pure gauge rank 2 tensor which is tangential and traceless (provided ð a φ a = 0), but no longer divergence-free. Theñ using (5.30), which expresses the gauge invariance of φ (2) .

Spin s modes and Young diagrams
As observed above, elements in C s can be identified with totally symmetric, traceless, divergence-free rank s tensor fields φ a 1 ...as on H 4 via It is useful to define also the mixed spin s objects, such as the "connection (2s−1)-form" which are all tangential and associated to the underlying irreducible rank s tensor field. Then the "field strength" form can be written as noting that the ðθ ... terms drop out for traceless tangential φ a 1 ...as , using (3.66). Here is some antisymmetrized derivatives corresponding to some two row rectangular Young projector P ∼ , which can be regarded as linearized higher-spin curvature. We will show below that the potential φ (s) a is then recovered from the following projection If the underlying φ a 1 ...as ∈ C 0 is an irrep of SO(4, 1), we may abbreviate this as Pure gauge modes. Finally, one can verify that a pure gauge rank s tensor drops out from the field strength form, As before, this is a manifestation of the gauge invariance of φ (s) . One way to see this is to move ∇ out of the brackets using (3.72) and, finally, use (D.4).
One may wonder about the meaning of the infinitesimal transformations These correspond to symplectomorphisms on CP 1,2 generated by the Hamiltonian vector field {Λ, ·}, which mix the different spin modes in a non-trivial way. They do not correspond to the above pure gauge modes (4.31), but see section 5.9.

Inner product and quadratic action
It is interesting and useful to compute the inner product (3.43) of the above spin s fields φ (s) defined by the trace in End(H). In the spin 1 case, consider the quadratic form which looks like the action for self-dual (abelian) Yang-Mills. In the spin 2 case, consider the analogous quadratic form because [φ (2) ] 0 = 0. Note that we used the symmetries (4.19) of R ae;bc or rather of its tangential part P R ae;bc , as the radial contributions drop out anyway. We observe that (4.35) is a (self-dual) linearized quadratic gravity action 13 , which can be written in terms of the R ab "forms" as follows: Similarly for spin s, we have which is again some self-dual quadratic Fronsdal-type higher-spin action [28]. The factor α s will be determined below. This suggests that a matrix model based on a single φ ∈ End(H) should define some higher-spin theory, which is however expected to be more or less trivial. Nevertheless it would be interesting to study the action defined by higher-order polynomials, and to understand its relation with Vasiliev's theory [29]. In the remainder of this paper, we will show how a non-trivial higher-spin gauge theory arises from multi-matrix models.

Projections, positivity and determination of α s
Now consider the spin s modes φ (s) ∈ C s as above, determined by some irreducible rank s tensor field on H 4 . We have seen that this in one-to-one correspondence to a spin s potential φ (s) a ∈ C s−1 as above. Then This provides the following relations: 13 The topological terms are the linearized Pontrjagin and Euler class (i.e. Gauss-Bonnet term).
Generic spin s case. In the generic case, we obtain similarly Explicit expressions for α s for s ≥ 3 could be computed similarly but are not required for our purposes.

Local decomposition
Finally consider any point on H 4 , for instance the reference point (2.6). We denote the four tangential coordinates with x µ , and the time-like coordinate on R 4,1 with x 0 . Then the so(4, 1) generators decompose (locally) into so(4) generators m µν , and the remaining translation generator by p µ = m µ0 . We can then decompose e.g. the spin s = 1 modes locally as and similar for higher-spin. From this point of view, the main lesson of the above results is that the φ µ (x) and φ µν (x) are not independent fields, but determined by the same irreducible spin 1 field φ a (x), and similarly for higher-spin fields. For generalized fuzzy spaces these constraints may disappear, as considered in [3]. For the basic spaces H 4 n and for S 4 N [2], the formalism developed above takes these constraints properly into account.

Matrix model realization and fluctuations
Now consider the IKKT matrix models with mass term, Here η ab = diag(−1, 1, . . . , 1) is interpreted as Minkowski metric of the target space R 1,D−1 . The positive mass µ 2 > 0 should ensure stability. The above model leads to the classical equations of motion plays the role of the Laplacian. Note that (5.2) are precisely the equation of motions for the IKKT model put forward in [30] after taking an IR cutoff into account.

Fuzzy H 4 n solution and tangential fluctuation modes
Consider the solution Y a = X a of (3.6) corresponding to fuzzy H 4 n , and add fluctuations on H 4 n . They naturally separate into tangential modes x a A a =0 and radial modes x a A a =0. The SO(4, 1)-invariant inner product is positive definite for (Hermitian) tangential A a on H 4 , and negative for the radial modes.
Since A a ∈ End(H n ) ⊗ R 5 , we expect four tangential fluctuation modes and one radial mode for each spin (except for spin 0), as for S 4 N [2]. Our strategy will be to remove the radial modes, and to find a useful basis of tangential modes in the semi-classical limit.
Intertwiners. Define the SO(4, 1) intertwiners They are Hermitian w.r.t. the inner product (5.5), and tangential except for I, noting that The SO(4, 1) Casimir for the fluctuation modes can be expressed using I as follows: is the vector generator of so(4, 1), and M (ad) bc = i{M bc , ·} denotes the representation of so(4, 1) induced by the Poisson structure on S 4 . As a check, we note that C (full) (x a ) = 0, since I(x a ) = 4x a . This reflects the full SO(4, 1)-invariance of the background x a .

Spin 0 modes
Let φ ∈ C 0 be a spin 0 scalar field. There are two tangential spin 0 mode, which read These modes satisfy a is a pure gauge field. Let us compute the action of the I intertwiner; to start with Similarly, one finds Now we can use the identities wherein we used for any tangential φ c , and the gauge fixing relations (5.30). Therefore For s = 0, this gives (2) , in agreement with (5.13). Then (5.17) gives a is tangential. To summarize, a (5.20)

Spin 1 modes
Now let φ a ∈ C 0 be a tangential, divergence-free spin 1 field. Then there are four tangential spin 1 modes, given by a ) will be computed for the general case below.

Spin 2 modes
Now let φ ab = φ ba ∈ C 0 be a tangential, divergence-free, traceless spin 2 field, and let φ Then there are four tangential spin 2 modes, given by Here φ (2) is the unique spin 2 mode in End(H), which involves the linearized Riemann tensor. They satisfy the gauge fixing relations derived below, see (5.30). Also recall from (4.15) that φ (2) c is indeed tangential. Furthermore, It is illuminating to display the explicit tensor content of the spin 2 modes, recalling that φ (2) b is the spin connection (4.13) and φ (2) is the curvature tensor. Using (4.17), this is (5.28) In particular, A a = θ ab A b encodes a so(4, 1)-valued gauge field A b = −ω b;de θ de given by the linearized spin connection of φ ab .

Spin s ≥ 1 modes.
Now consider the generic case. Let φ a 1 ...as ∈ C 0 be a tangential, divergence-free, traceless, symmetric spin s field, and let φ (s) Then there are four tangential spin s modes, given by Here φ (s) is the unique spin s mode in End(H). The modes (5.29) satisfy the gauge-fixing relations These relations hold for any spin. Together with (5.7), it follows that I(A (2) ) and I(A (3) ) are tangential, while I(A (1) ) and I(A (4) ) are not. Let us proceed toĨ; we first show that This can be proven inductively as follows: where the second term can be rewritten as So that we obtain for any spin s ≥ 1. Therefore On the other hand, for a spin s field φ we have for s ≥ 1, which is tangential. Hence if S 2 is diagonal then I is also diagonal, and the Casimir C 2 [SO(4, 1)] (5.8) can be diagonalized simultaneously. To evaluate (5.41),we decompose A a into its components in C s−1 ⊕ C s+1 as follows: where the last line is only a short-hand notation which applies to irreps, cf. (4.10). Hencẽ I is diagonalized as follows: This shows that all these modes are distinct, and it will allow to diagonalize and evaluate explicitly the quadratic action. It also implies that we did not miss any modes, since there can be only 5 modes for each spin (including the radial one, see below).
Gauge fixing term. The intertwiner G of (5.6) takes the values

Recombination, hs-valued gauge fields and Young diagrams
The distinct modes A (i) are useful to disentangle the different degrees of freedom. On the other hand we can relax the requirements that the underlying tensor fields φ a 1 ...as are irreducible, so that the modes can be captured in a simpler way.
which enters the four modes as follows Divergence modes. Now we drop the requirement that φ ab is divergence-free. Consider the case of rank 2 tensors, expressed in terms of spin 1 modes as in (4.22) Then according to (4.24), these contribution to the would-be spin 2 modes A a vanish identically. The contribution to A  15 . In particular, we need not worry about these constraints upon projecting H 4 to M 3,1 . It will suffice to impose the appropriate divergence-and traceconditions for M 3,1 . Finally as for S 4 N [2], we can collect all tangential fluctuation modes as hs-valued tangential gauge fields where A c,α (x) are double-traceless tensor fields corresponding to 2-row Young diagrams of the type . The external leg is associated to the extra box in the Young diagram. However the A c,α (x) are in fact higher curvatures of the underlying symmetric tensor fields φ a 1 ...as as in (4.28), which characterize the irreducible physical degrees of freedom A (i) a .

Inner products
The inner products (5.5) of the tangential fluctuations are given by Note that all A a modes are pure gauge modes, and they will drop out in the action.

Radial modes
Finally consider the radial fluctuation modes. These are given by They are dangerous because the radial metric in R 1,4 is negative, recalling that x a x a = −R 2 < 0. However, they disappear after the projection to M 3,1 . If we include these radial fluctuations, we should first diagonalize I. We have Recall that I(A (2 ,3) ) is tangential, but I (A (1,4) ) is not, with using (5.7) and (5.31). Hence the radial modes may couple to the A (1,2) or the B (1,2) modes, and the I eigenmodes seem to mix completely all 3 components A (1 ) , A (4) , A (r) . However since the radial modes are negative definite, we will focus on the tangential modes, and on its projection to M 4,1 in the next stage.

SO(4, 1)-invariant quadratic action on H 4
The quadratic fluctuations for the fluctuation modes y a = x a + A a are governed by the action Gauge-invariant action. Consider first the gauge-invariant kinetic term We verify that the pure gauge modes A a are null modes using (5.49) and (5.38): for any spin, taking into account the on-shell condition 1 2 µ 2 = 4r 2 . Hence the pure gauge modes A For spin 0, we determine the action explicitly for the B (1) and B (2) modes The inner product is diagonal for spin 0, and the quadratic action is given by Since B (1) a ∈ C 0 , this is indeed positive define (except for the pure gauge mode) due to (5.55), recalling that ∝ −ð · ð for spin 0 (3.69).
Gauge-fixed action and positivity. Now we consider a gauge-fixed action, which is obtained by canceling G with a suitable Faddeev-Popov (or BRST) term: We work in the basis {B (i) } (5.47) where I is diagonal. Then the eigenvalues of the kinetic operator D 2 are elaborated in the appendix D.1. Together with the inner products in section 5.2.1, we obtain the following diagonalized quadratic action 1 r 2 R 2 φ (s) (1 − α s ) + 2r 2 α s (s + 1) ( + 4r 2 (s + 1))φ (s) , (5.72a) All these terms are non-negative, because for any intertwiner A, using (D.33). The first line is positive because 1 > α s , and the second line is positive since + r 2 s(s − 3) is manifestly positive for s ≥ 3, while for s = 1, 2 it coincides with − 2r 2 which is also positive on divergence-free tensor fields as shown in (4.39). As usual, the unphysical modes will be canceled by Faddeev-Popov ghosts.
We consider explicitly the case of spin 1 and spin 2. For spin 1, we have and for spin 2 using (4.43). Note that we only include tangential fluctuation modes here. If we would also include the radial fluctuations as in section 5.3, they would be negative definite or ghost modes, because the metric in the radial direction is time-like. However this is resolved upon projecting to M 3,1 , as discussed below.

Yang-Mills gauge theory
We can write the full action (5.1) in a conventional (higher-spin) Yang-Mills form for the recombined higher-spin gauge fields (5.54) A a = θ ab A b . Then the field strength is recalling that ∇θ ab = 0. Hence the action (5.1) is basically a hs-valued Yang-Mills action 16 (dropping surface terms and using µ 2 = 8r 2 ), where is the a dimensionless Yang-Mills coupling constant. For nonabelian spin 1 modes A (4) a on stacks of H 4 n branes, the usual Yang-Mills action is recovered. For spin 2, one would expect this to describe some type of quadratic gravity action [31][32][33]. However this does not happen as shown below, since the graviton is obtained by a field redefinition (5.101) and does not propagate at the classical level. However the Yang-Mills framework suggests that no ghost modes appear also for higher-spin (as opposed to quadratic gravity), hence gravity might emerge at the quantum level.

Metric and gravitons on H 4
Now we take some of the leading (cubic) interactions of these modes into account, focusing on the contributions of the spin 2 (and spin 1) modes to the kinetic term on H 4 . These contributions are expected to give rise to linearized gravity. The kinetic term for all fluctuations on a given background Y a ∼ y a arises in the matrix model from 17 using (3.77); some dimensionful constants are absorbed in ϕ, and Greek indices indicate local coordinates. Here γ ab is a symmetric tensor in SO(4, 1) notation γ ab = η cc e ca e c b , e ca = {y c , x a } (5.80) which in local coordinates near some reference point ξ reduces to γ µν , cf. (3.58d). Hence the effective metric is given by [3,15,19] and e ca can be interpreted as vielbein. For a deformation of the H 4 background of the form Here H ab [A] is an SO(4, 1) intertwiner and tangential, Then the linearized effective metric (5.81) becomes in SO(4, 1)-covariant notation is dimensionless. We study the graviton modes (5.85) for the spin s = 0, 1, 2 fluctuations of (5.29) in more detail below.

Spin 0 gravitons
To begin with, consider the perturbation (5.83) of the metric for the two spin 0 modes of (5.11). One finds Upon averaging, one obtains Then the physical graviton of (5.85) satisfies the de Donder gauge, The spin 0 contribution to the metric is interesting because its off-shell modes have the wrong (ghost-like) sign in GR. This does not happen in the present Yang-Mills model, which is important for quantization.

Spin 1 gravitons
Next, we compute the spin one contributions to the gravitons on H 4 . Taking into account the C s gradation, the averaged metric perturbation (5.83) is non-vanishing only for the modes A a .

Spin 1 graviton
a . Here, we observe such that the averaging yields This has the form of pure gauge (diffeomorphism) contributions. Since the A (2) modes are pure gauge, they are not physical in the present model.
a . Similarly, we have such that averaging yields Physical spin 1 gravitons. For the spin 1 eigenmodes B (i) of (5.47), we therefore obtain the following physical gravitons:

(5.95)
Hence there is indeed a physical spin 1 mode h ab [B (3) ] contributing to the metric fluctuations. Nevertheless, since it has the form of pure gauge (diffeomorphism) contributions, it will decouple from a conserved energy-momentum tensor T µν .

Spin 2 gravitons
Finally, we consider the spin 2 fluctuations of the background and evaluate their associated gravition modes.
The second term drops out in the projection to C 0 , and using (4.29) twice one finds (5.97) In fact this is a pure gauge mode in the model.  In summary, the physical fields contributing to the metric fluctuations are a spin 2 field, a spin 1 field, and a spin 0 field. This is somewhat reminiscent of scalar-vector-tensor gravity. The spin 0 and spin 2 modes both satisfy the de Donder gauge.
To understand the present organization into spin modes, recall that the linearized metric fluctuations h ab decompose in general as ab is a divergence-free, traceless spin 2 tensor. This corresponds to our spin 2, spin 1 and spin 0 contribution to the graviton; note that ξ a contains another spin 0 (divergence) mode. While the ξ a fields are unphysical pure gauge modes, the spin 0 part h is a physical field which is in general sourced by the trace of the energy-momentum tensor. In the Einstein-Hilbert action, this spin 0 field enters with the "wrong" sign, cf. [34]. This does not happen here, which is certainly welcome for the quantization of the model.

Classical action for metric fluctuations
Having defined the notion of physical graviton in (5.85), an effective 4-dimensional action for h ab is desirable. By writing the trace as an integral as in (3.77), one can express the (gauge-fixed) kinetic term for B (4) in terms of h ab ≡ h ab [B (4) ] as follows: where g YM is the dimensionless Yang-Mills/Maxwell coupling constant (5.78). Superficially, this looks like a mass term for the graviton; however this is only the spin two mode, which is by definition invariant under diffeomorphisms. Hence (5.104) could also be viewed as the quadratic contribution to the cosmological constant in GR 18 . Taking into account a coupling to matter of the form δ h S = 1 2 h ab T ab , the equations of motion for h ab become Clearly h ab is not propagating, but acts like an auxiliary field which tracks T ab . As a consequence, the pure matrix model action (5.1) does not lead to gravity on H 4 , similar to the case of S 4 N [2]. Nevertheless, the action (5.1) does define a non-trivial, and apparently not pathological, spin 2 theory in 4 dimensions with a propagating spin 2 field φ ab , which should be suitable for quantization. Gravity may then arise upon quantization, as discussed next.

Induced gravity
At first sight it may seem disappointing that gravity does not arise from the classical action. On the other hand, since classical GR is not renormalizable, it should presumably be viewed as a low-energy effective theory. Adopting this point of view, it is reasonable that the starting point of an underlying quantum theory can be very different at the classical level, as for instance in the approach advocated here. This train of though is exactly the idea of emergent gravity 19 . As soon as quantum effects in the matrix model are taken into account, the effective metric h ab will unavoidably acquire a kinetic term, and therefore propagate. More specifically, it is well-known that induced gravity terms arise at one loop, upon integrating out fields that couple to the effective metric [17,18,37]. The induced terms include the cosmological constant and Einstein-Hilbert terms. The maximal supersymmetry of the underlying model 20 along with the finite density of states of the solution strongly suggests that the model is UV finite and "almost-local". Moreover, the usual large contribution to the cosmological constant √ gΛ 4 is avoided here, cf. the one-loop computation in [3]. Canceling also the induced Einstein-Hilbert term is more subtle 21 , and it is plausible that the supersymmetry breaking H 4 background does lead to an induced Einstein-Hilbert term with scalẽ Λ = O 1 r . Motivated by these considerations, one may add a term σΛ 2 h ab ð · ð h ab to the action (5.104), with σ = ±1, such that the total action coupled to matter reads The equation of motion for h ab are then whereΛ is the effective cutoff scale set by induced gravity. For σ = −1, this is indeed a reasonable equation for linearized gravity, with the effective Newton constant and mass scale The mass scale can become very small m 2 = O 1 and n is large, or upon projection to the Minkowski space-time M 3,1 , where the universe grows in time. Of course, the mass term will acquire quantum corrections too, which will be suppressed by supersymmetry. It would be desirable to study this in more detail elsewhere. Even though such a mass term might be interpreted in terms of a cosmological constant in linearized GR, its meaning here is somewhat different. As in GR, a proper interpretation requires the full non-linear theory. However, it is plausible that a positive mass term may simply imply an IR cutoff for gravity here, while the large-scale structure of the background solution might not be affected. Therefore a small, but non-zero mass term is quite welcome in the presented setting to ensure stability, while the large-scale cosmology would be determined by the background solution, as illustrated in section 6.

Local gauge transformations
Among the higher-spin gauge transformations δ Λ (x a + A a ) := {x a + A a , Λ} generated by Λ ∈ C, consider the spin 1 gauge transformations generated by with v a (x) a divergence-free vector field. These correspond to (volume-preserving) diffeomorphisms on H 4 . The action on scalar functions φ(x) reads so that the action on vector fluctuations is using (4.9). The first term describes a diffeomorphism corresponding to the vector field v a = α 1 ( − 2r 2 )v a . The second term accounts for the spin 1 pure gauge mode A (2 ) a as discussed in section 5.1, whose contribution to the graviton h ab was computed in (5.91). The higher-spin gauge transformations could be worked out similarly. Since there is only one such gauge invariance, but several fields for each spin, one may worry about the consistency of the model. However, recall that the gauge-fixed action (5.71) has been proven to be well-defined and non-degenerate in section 5.4. Hence there is no problem at least in the Euclidean setting. This is due to the special origin of the fluctuation modes in End(H n ), see (3.55).
6 Lorentzian quantum space-times from fuzzy H 4 n Having disentangled the fluctuations on H 4 n , we would like to apply these tools to the more interesting cosmological space-time solutions M 3,1 . Since the latter is obtained by a projection considered in section 3.2, many considerations remain valid. Most importantly, the fluctuation modes originate from the same End(H n ) such that we can rely on the same spin operator S 2 , and our classification can be carried over. Moreover, the tangential fluctuations on H 4 n are in one-to-one correspondence to the full set of fluctuation modes on M 3,1 , as will be shown below. The symmetry group is reduced to SO(3, 1) instead of SO(4, 1), which is weaker, but should still be very useful.

Cosmological space-time solutions
By projecting fuzzy H 4 n onto the 0123 plane via Π of (2.30) i.e. by keeping the Y µ = M µa α a for µ = 0, 1, 2, 3 and dropping Y 4 , we obtain (3+1)-dimensional fuzzy space-time solutions. Since the embedding metric η µν is compatible with SO(3, 1), we have Depending on α · α we obtain three different types of quantized space-time solutions with Minkowski signature in the IKKT model with mass term. These are: Choosing a positive mass term to ensure stability, we focus on the solution This is the homogeneous and isotropic quantized FLRW cosmological space-time M 3,1 n with k = −1 introduced 23 in [1]. Here m 2 sets the scale r 2 , while n remains undetermined. These backgrounds are SO(3, 1)-covariant, which is the symmetry respected by X .
Globally, we have the following constraints where η will be a global "cosmic" time coordinate. From the radial constraint x a x a = −R 2 on H 4 one deduces {x a x a , x µ } = 0, which further implies This establishes a relation between the momenta and the t µ , Furthermore, the self-duality constraint (3.58c) reduces to 24 where the last equation is simply a consequence of x µ t µ = 0. Therefore t µ describes a space-like S 2 with radius r −2 cosh 2 (η). Conversely, the above relations allow to express θ µν in terms of the momenta t µ as follows (6.10) By means of R ∼ 1 2 nr, one can summarize (6.10) neatly: where η µν α (x) is a SO(3, 1)-invariant tensor field on M 3,1 , which is analogs of the t'Hooft symbols. Note that θ 0i θ ij for late times η 1; this reflects the embedding of H 4 ⊂ R 4,1 which approaches the light cone at late times. Thus space is almost commutative, but space-time is not. Nevertheless the effects of non-commutativity will be weakened due to the averaging on S 2 . Finally the constraint (3.58d) reads which at the chosen reference point yields Averaging and effective metric on M 3,1 . An effective metric for scalar fields φ(x) on M 3,1 can be defined by the quadratic action (5.79). Looking at (6.12), we note that γ αβ contains the term t α t β , which is not constant on the fiber S 2 . By averaging over the fiber, one obtains the following result [1] [ (6.14) Note the signature change at cosh 2 (η) = 3 which marks the Big-Bang in this model, and the large pre-factors which grow in time η. Taking into account the conformal factor in the effective metric G µν (5.81), one obtains the cosmic scale parameter a(t) ∝ t for late times, corresponding to a coasting universe [1]. However we have not yet shown that this metric G µν governs all of the low-energy physics, and that there are no tachyonic or ghost modes. The large local symmetry of the model and the universal structure of the Yang-Mills action should help to elaborate the full dynamics.
Here we only take some steps in that direction: we establish a precise correspondence between the fluctuation modes as well as a close relation between the action of both spaces.

Wave-functions, higher-spin modes and constraints
In this section we briefly comment on the fluctuation modes on M 3,1 . The space of functions End(H n ) on M 3,1 is the same as on H 4 n , meaning that the decomposition (3.55) remains valid and is truncated at order n. The modes will still be considered as functions (or sections of higher-spin bundles) on H 4 , such that a representation as in (3.42) is expected to hold. Consequently, the modes can be interpreted as functions (or higher-spin modes) on 25 M 3,1 via Π of (2.30). The φ ab (x) etc. then define some higher-rank field on M 3,1 . In the following, we will only address a few basic points.
6.4 Tangential fluctuation modes, relation with H 4 and SO(4, 1) Now consider fluctuations y µ = x µ + A µ around M 3,1 . The first observation is that these four fluctuation modes A µ , µ = 0, . . . , 3 are in one-to-one correspondence with the tangential fluctuations on H 4 . To see this, recall that tangential fluctuations on H 4 satisfy by definition 26 the constraint can be mapped one-to-one to tensors h µν on M 3,1 , and the missing components h ab are uniquely determined from the h µν . In the same vein, all internal fluctuations on S 2 will be organized in a SO(4, 1)-covariant way as on H 4 . This relation is somewhat analogous to a Wick rotation.
Action and dynamics. The matrix model provides again an action for the fluctuation modes A µ , which has the same structure as in section 5.4, upon gauge fixing. The matrix Laplacian on M 3,1 is related to the one on H 4 through We can utilize the same mode expansion in terms of A (i) µ as in section 5.1, 20) which is SO(3, 1)-covariant. As explained in section 5.2, the irreducibility constraints, i.e. transversality and tracelessness, can be implemented as appropriate for M 3,1 without changing the setup. The relation (4.9) still applies; for example Note that ð c ð c is the Euclidean Laplace operator on H 4 , even though we are working in the Minkowski case. Hence the right-hand side of (6.21) amounts to some field redefinition.
In the same vein, the higher-derivative terms in the action (5.72) for the rank s tensor fields φ a 1 ...a 2 amount to field redefinitions. Therefore one should expect that these higherderivative terms do not lead to new degrees of freedom or ghosts.
On the other hand it might be tempting to use a SO(3, 1)-covariant formalism, where e.g.
µ }. However then some identities are lost, and it remains to be seen which formalism is more advantageous.

Conclusion and outlook
In this article we provide a careful and detailed analysis of the fluctuation modes on fuzzy H 4 n as a background in Yang-Mills matrix models, focusing mainly on the semi-classical case. While the results are largely analogous to the case of S 4 N [2], the present approach based on a suitable Poisson calculus is more transparent and fairly close to a standard field-theory treatment. The intrinsic structure of these quantum spaces is responsible for obtaining a higher-spin gauge theory, which is fully SO(4, 1)-covariant. The key feature is the equivariant bundle structure, which leads to a transmutation of would-be Kaluza-Klein modes into higher-spin modes.
Summary. Let us summarize the main points: A suitable set of representations for the construction of H 4 n is identified as the minireps or doubleton H n , for which we recall the oscillator realization in section 3. The first major step is a classification of the fuzzy algebra of functions End(H n ), which relies on two pillars: (i) the construction of a spin Casimir invariant S 2 which measures the intrinsic angular momentum on the S 2 n fiber, and (ii) the statement that the quantization map (3.32) is surjective. This provides the basis for the expansion (3.42) of a generic function in End(H n ), which is a finite expansion in the generators associated to the fiber. More precisely, End(H n ) decomposes into a finite set of higher-spin sectors C s (3.55), labeled by the spin Casimir. In the semi-classical limit, these become modules over the algebra of functions on H 4 , which are identified with tangential, divergence-tree, traceless rank s tensor fields on H 4 . The second major step is the development of a suitable differential calculus built upon derivations ð a defined via the Poisson bracket in (3.64). This provides the tools to work explicitly with the generic spin s modes on a non-compact space. Having in mind the IKKT matrix model, we observe that H 4 n is a solution of Yang-Mills matrix models with mass term, and classify the fluctuation modes around an H 4 n background in section 5. Building on the understanding of End(H n ), we find four tangential (5.29) and one radial fluctuation modes for each spin s ≥ 1. We find the explicit eigenmodes B (i) of the differential operator D 2 , see (5.65), which governs the fluctuations in the matrix model. It turns out that the tangential modes are stable, due to positivity results on their kinetic terms. Next, we identify the physical graviton (5.85) as linearized fluctuation of the effective metric (5.81) around the H 4 n background, and we compute the associated graviton modes for spin s = 0, 1, 2. The gravitons at spin s = 0, 2 naturally satisfy the de Donder gauge. However, it turns out that the spin 2 graviton behaves as an auxiliary field, at least at the classical level. A more interesting gravitational behavior should be obtained by including quantum corrections, leading to induced gravity terms. We briefly discuss this scenario in section 5.8. Considering H 4 n as a starting point towards the fuzzy space-time M 3,1 n , these issues are however less important. Since M 3,1 n is obtained from H 4 n by a projection, the fuzzy algebra of functions for M 3,1 n coincides with End(H n ), and our results provide a useful set of tools. As first steps, we briefly discuss the geometry and the organization of higher-spin modes of M 3,1 n , and establish a relation between tangential fluctuation on H 4 n and M 3,1 n in section 6.
Discussion and outlook. From a physics point of view, the results may seem a bit disappointing in the sense that the spin 2 modes do not lead to a propagating graviton at the classical level. However gravity could be restored in the quantum case, where induced gravity terms arise. The most encouraging result is that the tangential fluctuations are stable and do not lead to ghost-like modes. This is an improvement over GR where the off-shell conformal modes have the wrong sign, and arguably also over quadratic gravity where ghost-like modes arise at least superficially, cf. [32,33]. On the other hand, the radial modes are unstable here, which however could be cured by a radial constraint. There are several issues which deserve to be studied further. For example, the Poisson calculus developed here should be extended to the fully non-commutative case. Likewise, the relation of the present higher-spin gauge theory with Vasiliev theory should be clarified. Some of the structural statemens in sections 3.5 and 3.6 would deserve a more rigorous treatment. Furthermore, the 1-loop computation in [3] could easily be adapted, since H 4 is locally very similar to S 4 . This would allow to make more specific statements about the induced gravity terms, although to obtain the Einstein-Hilbert term may require a more refined approach. Finally, the minimal case n = 0 is very remarkable and special, because it does not correspond to a quantized symplectic space.
The main physics motivation for the present work is the close relation to the cosmological FLRW-type solutions M 3,1 of [1], which are obtained from a projection of H 4 n . The fluctuation analysis on M 3,1 can largely proceed along the same lines, with some important differences. In particular, the radial modes will disappear while the signature becomes Lorentzian. Furthermore, field redefinitions such as (5.101), which are responsible for the non-propagating nature of the graviton on H 4 , should no longer cancel the propagator. Therefore M 3,1 n is a very promising candidate for a quantum space-time with interesting gravitational physics in the framework of matrix models. However, we postpone a detailed analysis of M 3,1 to future work. with m, n, l = 1, 2, 3. They satisfy The U (1) E generator E = M 05 is the conformal Hamiltonian, whose spectrum is positive in a positive energy representation. Denoting the maximal compact Lie sub-algebra of SU (2) L ×SU (2) R ×U (1) E as L 0 , the conformal algebra g has a three-graded decomposition with respect to E, such that and L ± are the non-compact generators. The six roots of so(6) C decompose accordingly into two compact roots X ± β i and four non-compact roots X ±α ij . The latter transform as (2) L ⊗ (2) R i.e. as complex vectors of SO(4), and satisfy (X ±α ij ) † = −X ∓α ij . Spinorial representations of SO(4, 2) are obtained in terms of the SO(3, 1) gamma matrices γ µ satisfying {γ µ , γ ν } = −2η µν for µ, ν = 0, 1, 2, 3 and γ 4 := γ 0 γ 1 γ 2 γ 3 as follows 27 : We adopt the gamma matrix conventions where σ m , m = 1, 2, 3 are the usual Pauli matrices. They satisfy as it should be. The universal covering group of SO(4, 2) is SU (2, 2), which is the group of 4 × 4 complex matrices with which respects the indefinite sesquilinear form The 15-dimensional Lie algebra su(2, 2) = so(4, 2) can thus be identified with the space of traceless complex 4 × 4 matrices Z α β with real structure

C.1 Functions on H n , spin Casimir S 2 and quantization
First we argue that any "reasonable" operator Φ ∈ End(H n ) (in particular any Hilbert-Schmidt (HS) operator) can be written as quantization (3.32) of some (square-integrable, in the HS case) function φ(m) on CP 1,2 . To see this, assume that some operator A is orthogonal to the space spanned by (C.6), i.e. Tr(AΦ) = 0 for all Φ as above. Then for all (square-integrable, say) functions φ(m), and therefore i.e. the symbol of A vanishes. It is well-known that then A also vanishes, cf. [42]; indeed m |A|m is holomorphic in m and m, and therefore vanishes identically if it vanishes on the "diagonal" m| A |m . The point is that coherent states |m are holomorphic in m. Now consider the spin operator (3.44) be the non-compact root generators. Then Now let |0 be the ground state of H n , which satisfies =: S 2 n |0 (C.14) using C 2 [so(4)]|0 = 2 n 2 n 2 + 1 |0 and E |0 = 1 + n 2 |0 , see (3.28). Now consider noting that the cross-terms cancel. Then Z − j |0 = 0 = 0|Z + j , and for the minimal case H 0 we have moreover M ij |0 = 0 and E|0 = |0 . We conclude The same argument applies for any point ξ ∈ H 4 , and (C.6) implies that End(H 0 ) contains only spin S 2 = 0 states. For H n with n ≥ 1, we have to consider the entire SO(4) orbit g |0 0| = g · |0 0| · g =: |m m| over ξ where |m = g · |0 for g ∈ SO(4). We can express M ij |m m|M ij in terms of the SO(4) Casimir by SO(4) invariance. The same argument applies for any point ξ ∈ H 4 in (C.6). Therefore S 2 decomposes End(H n ) into spin s irreps as follows S 2 End(Hn) = n s=0 2S 2 n + 2s(s + 1)1l s − n(n + 2) + 2( 2s(s + 1)1l s (C.18) which implies (3.55) and (3.56). This respects the structure of a C 0 module, hence C can be viewed as a bundle over H 4 with fiber given by the space of functions on the fuzzy sphere S 2 n , with all multiplicities equal to one.

C.2 Minimal fuzzy H n=0
Consider the minireps (3.28) for the minimal case n = 0 = j L = j R . Then the lowest weight state |Ω = |1, 0, 0 of H 0 is the unique eigenspace with eigenvalue of E = 1, and there are only four non-vanishing operators with L + |Ω = a i b j |Ω = 0, while the SO(4) generators vanish on Θ ij |Ω = 0, i, j = 1, . . . , 4. Hence the X a generate a 4-dimensional quantized hyperboloid H 4 0 without extra fiber. This does not seem to correspond to a coadjoint orbits of SO(4, 2), and the coherent states SO(4, 2)|Ω form a trivial U (1)-bundle over H 4 . Consider the structure of H 0 in some detail. Then spec(E 0 ) = {1, 2, 3, . . .}, and the subspace for each given eigenspace of E 0 has the structure . . , 4 (C.10) link the neighboring sub-spaces of (C. 19) with E 0 and E 0 ± 1, and similarly the Θ 0j = r 2 . On the other hand, the Θ ij are the SO(4) generators acting within (C. 19). Therefore Ω|Θ ab |Ω = 0 vanishes for the local coherent states, but nevertheless Θ ab = 0 as operator. Hence H 0 is a non-commutative space which is not the quantization of a symplectic space. This is very interesting and should be investigated in more detail elsewhere. For some related mathematical results see [43]. The structure of H n with n ∈ N is analogous, where (C.19) is replaced by D Auxiliary identities for semi-classical H 4 n For any tangential φ a ∈ C, the following identity holds: For any tangential, divergence-free φ a ∈ C 0 , formula (3.67) yields Moreover, one can verify for any φ that for any φ ∈ C, and for any scalar function f ∈ C 0 , using (D.3). Finally, For the reducible tensor contributions (4.23), we need Covariant derivative ∇. Recalling ∇ a ∇ c φ = P cc ð a ð c φ, we obtain = − 1 r 2 R 2 {θ ac , φ} − r 2 (x a ð c − x c ð a )φ = − 1 r 2 R 2 P aa P cc {θ a c , φ} = P aa P cc [ð a , ð c ]φ (D.10) Hence, the ∇ a commute on scalar functions. For generic tensor fields φ b 1 ...bn we have to be more careful and proceed as follows: ∇ a ∇ c φ b 1 ...bn = ∇ a P b 1 b 1 · · · P bnb n ð c φ b 1 ...b n = P cc P b 1 b 1 · · · P bnb n ð a P b 1 b 1 · · · P b n b n ð c φ b 1 ...b n (D.11) = P cc P b 1 b 1 · · · P bnb n ð a ð c φ b 1 ...b n + P b 1 b 1 · · · P bnb n ð a P b 1 b 1 · · · P b n b n ð c φ b 1 ...b n Inspecting the second term in more detail, we arrive at Consequently, the commutator looks as follows: With a little relabeling, we obtain P ab j P cd − P cb j P ad φ b 1 ...b j−1 db j+1 ...bn (D.14) ≡ R ac φ b 1 ...bn .
Identities for spin s fields. The following identity holds for any tangential traceless divergence-free spin s field φ a ∈ C: and x a ð a φ = 0. Here ð · ð is the Euclidean Laplacian on H 4 . Further, using the self-duality we have the identity which is (2.22). Note that (D.24) holds for any divergence-free, tangential φ b ∈ C.
Graviton identity The following identity will be useful for tangential A a , using (D.4) and (5.7); note that ð a respects the projection [·] 0 .

D.1 Casimirs, positivity, and eigenvalues of D 2
In order to show that the kinetic term is positive, we need some positivity results. A first result for spin 1 is the following. Assume that φ (1) is Hermitian and determined by the tangential divergence-free vector field φ a as in (4.7). Then This implies that ( − 2r 2 )φ a is positive for divergence-free square-integrable tangential tensor fields, cf. (4.39). In particular, this giveŝ We also observe φ (s) φ (s) ∝ ð a φ (s) ð a φ (s) ≥ 0, φ (s) ∈ C s (D. 29) using (3.78), since x·ð = 0, i.e. ðφ has no radial components, hence the metric is Euclidean. Therefore is a positive operator on any square-integrable φ ∈ C s . For higher spin, we need the following intertwining property of the vector fluctuations