]>NUPHB14587S0550-3213(19)30061-610.1016/j.nuclphysb.2019.02.027The Author(s)High Energy Physics – TheoryFig. 1Sketch of the projection Πx from H4 to M3,1 with Minkowski signature.Fig. 1The fuzzy 4-hyperboloid Hn4 and higher-spin in Yang–Mills matrix modelsMarcusSperling⁎marcus.sperling@univie.ac.atHarold C.Steinackerharold.steinacker@univie.ac.atFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, AustriaFaculty of PhysicsUniversity of ViennaBoltzmanngasse 5ViennaA-1090Austria⁎Corresponding author.Editor: Leonardo RastelliAbstractWe consider the SO(4,1)-covariant fuzzy hyperboloid Hn4 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 H4 taking values in a higher-spin algebra associated to 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 Hn4. We establish a one-to-one correspondence between the tangential fluctuations of these spaces.1IntroductionIn 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 Rθ4 [4,5] are not fully covariant, as an explicit tensor θμν breaks the covariance. In previous work [2,3], gauge theory on the fuzzy 4-sphere SN4 was studied in detail, starting from the observation that SN4 is a solution of Yang–Mills matrix models supplemented by a mass term, cf. [6]. Here we extend this analysis to fuzzy Hn4, which is a non-compact quantum space preserving an SO(4,1) isometry, also known as Euclidean AdS4. For other related work on covariant quantum spaces see e.g. [7–14].The motivation for this work is two-fold: first, we want to develop a formalism to study gauge theory on Hn4 along the lines of usual calculus and field theory, in order to facilitate the interpretation of the resulting models. While SN4 allows to use a clean but less intuitive organization of fields into polynomials corresponding to Young diagrams, the non-compact nature of H4 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 Mn3,1 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. Mn3,1 can obtained from the present Hn4 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 Hn4. We establish the relevant formalism in this paper, and moreover provide some explicit links between the modes on Hn4 and Mn3,1.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 S2-bundles over the base space (i.e. S4 or H4 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 H4, we develop a calculus, solely based on the Poisson structure, to organize the fuzzy algebra of functions on Hn4 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 H4.Having understood the “functions” on Hn4, we proceed by considering Hn4 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 SN4 [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 M3,1. We will establish a one-to-one correspondence of the tangential modes on Hn4 to the full set of fluctuations on M3,1. Since the tangential modes on Hn4 are stable and free of pathologies (in contrast to off-shell GR), it seems likely that the Minkowski setting on M3,1 provides a good model, too. In fact, the presence of negative radial modes on Hn4 would require to implement a constraint in the matrix model, which may spoil supersymmetry. This is not needed for M3,1, which provides further motivation for including a discussion of M3,1 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 Hn4 in section 2, before discussing fuzzy Hn4 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 Hn4, we explore the fluctuations around an Hn4 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 graviton modes are identified and their equation of motions are derived. Before concluding we briefly explore the projection of Hn4 to the Minkowskian Mn3,1 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.2Classical geometry underlying Hn4The classical geometry underlying fuzzy Hn4 is CP1,2, which is an S2-bundle over the 4-hyperboloid H4. More precisely, CP1,2 is an SO(4,1)-equivariant bundle over H4 as well as a coadjoint orbit of SO(4,2). Recall, for instance from [20, Def. 1.5], that a G-equivariant bundle π:E→X is equipped with a G-group action ρ˜:E→E as well as ρ:X→X such that the projection map π is an intertwiner, i.e. π∘ρ˜=ρ∘π. Here, the actions of SO(4,1) on the total space CP1,2 and base space H4 are immanent by definition of these spaces. In particular, this means that the local stabilizer group SO(4) acts non-trivially on the fiber S2, leading to higher-spin fields on H4, and a canonical quantization exists. The construction is similar to twistor constructions for Minkowski space.2.1CP1,2 as SO(4,1)-equivariant bundle over the hyperboloid H4Let ψ∈C4 be a spinor of so(4,1) with ψ¯ψ=1. Consider the following Hopf map:(2.1)H4,3→H4⊂R1,4ψ↦xa=r2ψ¯γaψ,a=0,1,2,3,4, where r introduces a length scale, and H4,3 is the 7-hyperboloid(2.2)H4,3={ψ∈C4|ψ¯ψ=ψ†γ0ψ=1}. The γa, a=0,…,4 are SO(4,1) gamma matrices, see appendix B for details. The map (2.1) is a non-compact version of the Hopf map S7→S4, which respects SO(4,1) and in which the xa transform as SO(4,1) vectors. By using (B.3) one can verify that(2.3)∑a,b=04ηabxaxb=−r24=:−R2 so that the right-hand side is indeed in H4; note that xa∈R due to (A.7). Since the overall phase of ψ drops out, we can re-interpret (2.1) as a map(2.4)xa:CP1,2→H4⊂R1,4 where CP1,2=H4,3/U(1) is defined as space of unit spinors ψ¯ψ=1 modulo U(1). In other words, CP1,2 is a S2-bundle over H4. To exhibit the fiber, consider an arbitrary spinor ψ with ψ¯ψ=1. Since(2.5)x0=r2ψ†ψ>0, there exists a suitable SO(4,1) transformation such that(2.6)xa|ξ=R(1,0,0,0,0), which defines a reference point ξ∈H4. Its stabilizer group is(2.7)H={h;[h,γ0]=0}=SU(2)R×SU(2)L⊂SO(4,1) where SU(2)L acts on the +1 eigenspace of γ0. By introducing complex parameters(2.8)ψT=(a1⁎,a2⁎,b1,b2),1=ψ¯ψ=−|a1|2−|a2|2+|b1|2+|b2|2=ψ†ψ it follows that |b1|2+|b2|2=1 and a1=a2=0. Thus after an appropriate SU(2)L transformation we can assume(2.9)ψT=(0,0,0,1), which will be a reference spinor over ξ throughout the remainder. Hence CP1,2 is a S2-bundle over H4, and the S2 fiber is obtained by acting with SU(2)L on ψ. This is analogous to the well-known fact that CP3 is an S2-bundle over S4. Note that the metric on the hyperboloid induced via(2.10)xa:H4↪R1,4 is Euclidean, despite the SO(4,1) metric on target space. This is obvious at the point ξ=(R,0,0,0,0), where the tangent space is R12344.SO(4,2) formulation and embedding functions. It is useful to view CP1,2 as a 6-dimensional coadjoint orbit of SU(2,2)(2.11)CP1,2≅{U−1ZU,U∈SU(2,2)}↪su(2,2) through the rank one 4×4 matrix(2.12)Z=ψψ¯,Z2=Z,tr(Z)=1,Z†=γ0Zγ0−1. The embedding (2.11) is described by the embedding functions(2.13)mab=tr(ZΣab)=ψ‾Σabψ=(mab)⁎,xa=rtr(ZΣa5)=r2ψ‾γaψ=rma5,a,b=0,…,4 noting that 12γa=Σa5, see (A.5). Upon restricting to so(4,1)⊂so(4,2)≅su(2,2), we recover (2.4), which reflects that the SO(4,1) action is transitive on CP1,2. The last equation in (2.13) amounts to a group-theoretical definition of the Hopf map, which will generalize to the non-commutative case. The SO(4,2) structure is often useful, but it does not respect the projection to H4.We can compute the invariant functions(2.14)∑0≤a<b≤4mabmab=∑0≤a<b≤4ψ¯ψ¯Σab⊗Σabψψ=12,(2.15)∑0≤a<b≤5mabmab=∑0≤a<b≤5ψ¯ψ¯Σab⊗Σabψψ=34, using the identities (B.6) and (B.7). Here, the indices are raised and lowered with ηab=diag(−1,1,1,1,1,−1). Combining the two identities (B.6)–(B.7) and recalling xa=rma5, we recover(2.16)xaxa=−r24=−R2. Remarkably, the SO(4,1)-invariant xaxa is constant on CP1,2. Similarly, (B.9) together with the above relations imply11This is just a manifestation of the relation Z2=Z, see (2.12). the SO(4,2) identities(2.17)ηcc′macmbc′=14ηab,a,b=0,…,5 which reduces to the SO(4,1) relation(2.18)ηcc′macmbc′−r−2xaxb=14ηab,a,b=0,…,4. In particular, this implies that mab is orthogonal to xa,(2.19)xamab=0. Furthermore, the following SO(4,2) identities hold:(2.20)ϵabcdefmabmcd=ψ¯ψ¯ϵabcdefΣab⊗Σcdψψ=2ψ¯ψ¯(Σef⊗1+1⊗Σef)ψψ(2.21)=4ψ¯Σefψ=4mef, using (B.10); this can also be seen from (B.11). Reduced to SO(4,1), this implies(2.22)ϵabcdemabmcd=−4rxe,e=0,…,4. Finally, there exists a self-duality relation(2.23)ϵabcdemabxc=ψ¯ψ¯ϵabc5deΣab⊗Σc5ψψ=12ψ¯ψ¯(Σde⊗1+1⊗Σde)ψψ(2.24)=ψ¯Σdeψ=mde using (B.10). Thus mab is a tangential self-dual rank 2 tensor on H4, in complete analogy to SN4 [21]. At the reference point (2.6), one can express mab in terms of the SO(4) t'Hooft symbols(2.25)mμν=ημνiJi,JiJi=1 where Ji describes the internal S2. This exhibits the structure of CP1,2 is an SO(4,1)-equivariant bundle over H4. The fiber S2 is generated by the local SU(2)L, while SU(2)R acts trivially.2.2CP1,2 as SO(3,2)-equivariant bundle over the hyperboloid H2,2Equivalently, the homogeneous space CP1,2 of SO(4,2) can be viewed as S2-bundle over H2,2, which arises from a different Hopf map(2.26)H4,3→CP1,2→H2,2⊂R2,3 as follows, cf. [22]:(2.27)ta=1Rψ‾Σa4ψ=1Rma4,a=0,1,2,3,5. This map is compatible with SO(3,2), and establishes (2.26) as SO(3,2)-equivariant bundle in the aforementioned sense. The reference spinor (2.9) is now projected to ta=r−1(0,0,0,0,1)∈R3,2, which transforms as SO(3,2) vector. Then ta defines a hyperboloid H2,2⊂R3,2 with intrinsic signature (+,+,−,−). Using analogous identities as before, we obtain the constraints(2.28)η˜abtatb=r−2,η˜ab=diag(−1,1,1,1,−1),taxa=0=tμxμ. The last relation follows from the SO(4,2) relation (2.17), noting that t4≡0. More generally, we can consider(2.29)xa=mabαb,ta=mabβb where α,β∈R2,4 are two linearly independent vectors with22The case of light-like α is also interesting, see section 3.2. αbβb=0. Then the previous constructions are recovered for α=e5,β=e4. The common symmetry group which preserves both αb and βb is SO(3,1). Note that t5∝x4 on CP1,2.2.3SO(3,1)-invariant projections and Minkowski signatureSo far we have constructed H4 and H2,2, 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(2.30)Πx:CP1,2→R3,1,m↦xμ=mμbαbΠt:CP1,2→R3,1,m↦tμ=mμbβbwith μ=0,1,2,3, which respect SO(3,1). A sketch of Πx is displayed in Fig. 1. In section 6, the image M3,1⊂R3,1 of Πx serves as cosmological FLRW space-time with k=−1, as discussed in [1]. In contrast, tμ is interpreted as internal space related to translations.3The fuzzy hyperboloid Hn4Now we turn to the central object of this paper: the fuzzy hyperboloid Hn4. Hn4 is a quantization of the bundle CP1,2 over H4, which respects the SO(4,2) structure and the projection to the base space H4. This is natural because CP1,2 is a coadjoint orbit of SO(4,2) via (2.11). As such CP1,2 is equipped with a canonical SO(4,2)-invariant Poisson (symplectic) structure; whereas on H4 no such structure exists. Hn4 was first discussed in [22], and it serves as starting point for a quantized cosmological space-time in [15].As for any coadjoint orbit, fuzzy Hn4 can be defined in terms of the operator algebra End(Hn), where Hn is a suitable unitary irrep of SU(2,2)≅SO(4,2). The representation is chosen such that the Lie algebra generators Mab∈End(Hn) generate a non-commutative algebra of functions, interpreted as quantized or fuzzy CPn1,2. The Mab are naturally viewed as quantized coordinate functions mab (2.13) on CP1,2. Fuzzy Hn4 is then generated by Hermitian generators Xa∼xa, 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 Hn4 explicitly, let ηab=diag(−1,1,1,1,1,−1) be the invariant metric of SO(4,2), and let Mab be the Hermitian generators of SO(4,2), which satisfy(3.1)[Mab,Mcd]=i(ηacMbd−ηadMbc−ηbcMad+ηbdMac). We choose a particular type of (discrete series) positive-energy unitary irreps33Strictly speaking there are two versions HnL or HnR with opposite “chirality”, but this distinction is irrelevant in the present paper and therefore dropped. Hn known as minireps or doubletons [23,24]. Remarkably, the Hn remain irreducible44This follows from the minimal oscillator construction of Hn, where all SO(4,2) weight multiplicities are at most one, cf. [23,25,26]. under SO(4,1)⊂SO(4,2). Moreover, the minireps have positive discrete spectrum(3.2)spec(M05)={E0,E0+1,…},E0=1+n2 where the eigenspace with lowest eigenvalue of M05 is an n+1-dimensional irreducible representation of either SU(2)L or SU(2)R. Then the Hermitian generators(3.3)Xa≔rMa5,a=0,…,4[Xa,Xb]=−ir2Mab=:iΘab (note the signs!) transform as SO(4,1) vectors, i.e.(3.4) Because the restriction to SO(4,1)⊂SO(4,2) is irreducible, it follows that the Xa live on a hyperboloid,(3.5) with some R2 to be determined below. Since X0=rM05>0 has positive spectrum, this describes a one-sided hyperboloid in R1,4, denoted as Hn4. Analogous to fuzzy SN4, the semi-classical geometry underlying Hn4 is CP1,2 [22], which is an S2-bundle over H4 carrying a canonical symplectic structure. In the fuzzy case, this fiber is a fuzzy 2-sphere Sn2. We work again in the semi-classical limit. We also note the following commutation relations(3.6)□XXb=[Xa,[Xa,Xb]]=−4r2Xb. The negative sign arises from η=diag(−1,1,1,1,1,−1), and □X is not positive definite.3.1Fuzzy Hn2,2 and momentum spaceAs in the classical case (2.27) and for later purpose, we also define(3.7)Ta=1RMa4,a=0,…,3,5 where RrT5=−X4. As the restriction of Hn to SO(3,2)⊂SO(4,2) is irreducible, the operators (3.7) satisfy the constraint(3.8)η˜abTaTb=−T0T0+∑i=1,2,3TiTi−T5T5=1r21 cf. (2.28). This is the quantization of the hyperboloid H2,2⊂R3,2 with intrinsic signature (+,+,−,−) of section 2.2 and becomes Lorentzian via the projection (2.3). The commutation relations are(3.9)[Ta,Tb]=i1R2Maba,b=0,…,3,5,[Tμ,Xν]=i1RημνX4,μ,ν=0,…,3, which justifies to consider Tμ as translation generators, and(3.10)□TTb=[Ta,[Ta,Tb]]=+4R2Tb Note the different signs in (3.10) and (3.6), which arise from of η55=−1=−η44.3.2SO(3,1)-covariant fuzzy spacesIn analogy to section 2.3, we consider the SO(3,1)-covariant fuzzy generators(3.11)X˜μ=Mμaαa,T˜μ=Mμaβa where α,β are SO(3,1)-invariant. They satisfy(3.12)[X˜μ,X˜ν]=(α⋅α)Mμν,[T˜μ,T˜ν]=(β⋅β)Mμν[X˜μ,T˜ν]=i(δμνMabαaβb+α⋅βMμν)=i(δνμα∧βD+α⋅βMμν) where α∧β=α4β5−α5β4 and D=M45. For α⋅α≈0 and α⋅β≈1≈α∧β, the X˜μ become almost commutative and the commutation relations are not far from the Poincare algebra:Poincare algebra. In particular for light-like α=12(1,−1) and β=12(1,1), we obtain(3.13)Kμ≔12(Mμ5−Mμ4),T˜μ=12(Mμ5+Mμ4) which satisfy(3.14)[T˜μ,T˜ν]=0=[Kμ,Kν],[T˜μ,Kν]=i(δνμD+Mμν). Hence the T˜μ together with Mμν generate the Poincare algebra ISO(3,1) as sub-algebra of so(4,2), with special conformal generators Kμ and the dilatation operator D(3.15)[D,T˜μ]=iT˜μ,[D,Kμ]=−iKμ.3.3Oscillator realization, minireps and coherent statesThe Hilbert space Hn is a highest-weight unitary representation of SU(2,2), which can be obtained by quantizing the spinorial construction of CP2,1 in (2.1). For the quantization one replaces the classical 4-component spinor ψα by 4 operators, which satisfy(3.16)[ψα,ψ¯β]=δαβ. The associated bilinears(3.17)Mab≔ψ¯Σabψ realize the Lie algebra (3.1) of SO(4,2), due to(3.18)[ψ¯Σabψ,ψ¯Σcdψ]=ψ¯[Σab,Σcd]ψ. The Mab are self-adjoint operators, since(3.19)Σab†=γ0Σabγ0−1. 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 HΛ.The oscillator algebra (3.16) can be realized explicitly as follows (cf. [24,27]): Consider bosonic creation and annihilation operators ai,bj which satisfy(3.20)[ai,aj†]=δij,[bi,bj†]=δijfor i,j=1,2. Using the ai,bj we form spinorial operators(3.21)ψ≔(a1†a2†b1b2) with Dirac conjugates(3.22)ψ¯≡ψ†γ0=(−a1,−a2,b1†,b2†). Then(3.23)[ψα,ψ¯β]=δβα as required, and the SO(4,2) generators are(3.24)Mab=ψ¯Σabψ=(−a1,−a2,b1†,b2†)Σab(a1†a2†b1b2). The generators of SU(2)L and SU(2)R are defined byLik≔ak†ai−12δikNaRji≔bi†bj−12δjiNb and the time-like generator X0 (or the “conformal Hamiltonian” E) is given by(3.25)r−1X0=E=M05=ψ¯Σ05ψ=12ψ†ψ=12(Na+Nb+2), where Na≡ai†ai, Nb≡bj†bj are the bosonic number operators, and(3.26)Nˆ=ψ¯ψ=−Na+Nb−2 is invariant. The non-compact generators are given by linear combinations of creation and annihilation operators of the form ai†bj† and aibj.Minireps. The simplest class of unitary representation has lowest weight space given by the Fock vacuum ai|0〉=0=bi|0〉, which defines [27](3.27)|Ω〉≔|1,0,0〉=:|0〉,E=1,jL=jR=0. This gives the doubleton minireps built on the lowest weight vectors(3.28)|Ω〉≔|E,n2,0〉≔ai1†…ain†|0〉,E=1+n2,jL=n2,jR=0|Ω〉≔|E,0,n2〉≔bi1†…bin†|0〉,E=1+n2,jL=0,jR=n2 which are annihilated by all L− operators, i.e. of the form aibj,(3.29)aibj|Ω〉≡0 and(3.30)n2≔(Nˆ+2)2=(Na−Nb)2,n=0,1,2,…. Acting with all operators of the form ai†b†j of L+ on |Ω〉, one obtains positive energy discrete series UIR's HΛ of U(2,2) with lowest weight Λ=(E,n2,0) and Λ=(E,0,n2). We will largely ignore the distinction and denote both as Hn. These are known as minireps of so(4,2), because they are free of multiplicities in weight space.55This can be seen e.g. from the characters given in [26]. They correspond to fields living on the boundary of AdS5. The minireps remain irreducible under SO(4,1) as well as SO(3,2), and they can be interpreted as massless fields on AdS4, or as conformal fields66It may seem tempting to apply some of the standard technology of CFT in the present context. However, the use of SO(4,2) here is quite different from CFT, and it does not respect the bundle structure over H4. Also, the notions of primaries and descendants do not seem to be applicable here, since in the present signature Kμ (3.15) do not rise or lower the eigenvalues of D. on Minkowski space. The lowest weight state |E,0,n2〉 of Hn generates a (n+1)-dimensional irreducible representation of either SU(2)L or SU(2)R with degenerate X0, naturally interpreted as fuzzy Sn2.Comparing the above oscillator construction (3.17) with (2.13), it is manifest that for each Hn, with n>0, the Mab generators can be interpreted as quantized embedding functions(3.31)Mab∼mab:CP1,2→so(4,2)≅R15. This provides the quantization of the coadjoint orbits (2.11), which defines fuzzy CPn1,2. Since X0≥1, they should be viewed as quantized bundles with base space Hn4 described by Xa, and fiber Sn2, 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 [16,22], 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 Hn provide a natural definition of coherent states |m〉=g⋅|Ω〉∈Hn, 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 CP1,2, and allow to recover the semi-classical geometry of CP2 as S2-bundle over H4 via mab=〈m|Mab|m〉. In particular, the lowest weight state is located at the reference point 〈Ω|Xa|Ω〉=xξa=(R,0,0,0,0), see (2.6). The local SO(4) generators Mij act on the coherent states over ξ in a spin n2 irrep.These coherent states |m〉 also provide a SO(4,2)-equivariant quantization map from the classical space of functions on CP1,2 to the fuzzy functions End(Hn):(3.32)Q:C(CP1,2)→End(Hn)f(m)↦∫CP1,2dμf(m)|m〉〈m| where |m〉 is a coherent state,77Observe that the phase ambiguity of the coherent states drops out here. and dμ is the SO(4,2)-invariant measure. For polynomial functions, this corresponds to Weyl quantization, mapping irreducible polynomials P(mab) to the corresponding totally symmetrized polynomials P(Mab); in particular Q(mab)=Mab. Likewise, square-integrable functions on CP1,2 are mapped to Hilbert-Schmidt operators in End(Hn). We expect88For a formal argument see appendix C.1. A more rigorous proof would be desirable. that the map Q is surjective, and that all “reasonable” (e.g. square-integrable or Hilbert-Schmidt) harmonics in End(Hn) can be obtained as quantizations of higher-spin harmonics on H4 via Q. This will be used below.3.4Algebraic properties of fuzzy Hn4Using the aforementioned oscillator realization, one can derive a number of useful identities for the above operators on Hn; we refer the reader to appendix C for the details. To begin with, consider the SO(4,1)-invariant radius operator(3.33)R2≔∑a,b=0,1,2,3,4ηabXaXb. Since HΛ is irreducible under so(4,1), it must follow that R2∼1. Indeed, one finds(3.34)XaXa=−r24Nˆ(Nˆ+4)=−r24(n2−4)=:−R2 where n=|Nˆ+2|=0,1,2,…. Note that R2 is positive for n=0,1, which seems strange because X0 is positive. However, this is a quantum artifact, and the expectation values 〈Xa〉 under coherent states still sweep out the usual H4. Additionally we compute the quadratic SO(4,1) and SO(4,2) Casimir operators(3.35)C2[so(4,1)]=∑a<b≤4MabMab=12(n2−4),(3.36)C2[so(4,2)]=∑a<b≤5MabMab=34(n2−4). We note that (3.35) agrees with [22]. Further identities can be obtained from the so(6)C identity (B.9), which entails(3.37)ηcc′MacMbc′+(a↔b)=12(n2−4)ηab. This implies the so(4,1) relation(3.38)ηcc′ΘacΘbc′+(a↔b)=r2(2R2ηab+(XaXb+XbXa)). These correspond to (2.17), (2.18). Moreover, one finds(3.39)XbMab+MabXb=0, which means that the SO(4,1) generators Mab are tangential to Hn4. Another interesting identity is(3.40)ϵabcdefMabMcd=4nMefϵabcdeMabMcd=4nr−1Xe cf. (2.21), (2.22). Finally, the self-duality relation (2.24) becomes(3.41)ϵabcdeMabXc=nrMde. To summarize, we have found counterparts for all relation of the classical geometry in section 2.1, which vindicates the choice of representation Hn.3.5Wave-functions and spin CasimirGiven a representation Hn of SO(4,2), the most general “function” in End(Hn) can always be expanded as follows(3.42)ϕ=ϕ(X)+ϕab(X)Mab+…∈End(Hn)=:C, which transform in the adjoint representation Mab↦[Mab,⋅] of so(4,2). The ϕab(X) will be interpreted as quantized tensor fields on H4, which transform under SO(4,1). We define an SO(4,2)-invariant inner product on C via(3.43)〈ϕ,ψ〉=trH(ϕ†ψ). For polynomials generated by the Xa, this trace diverges. However this is only an IR-divergence, 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 Mab.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(3.44)S2≔C2[so(4,1)]+r−2□=∑a<b≤4[Mab,[Mab,⋅]]+r−2[Xa,[Xa,⋅]], which measures the spin along the S2 fiber. To understand this, we locally decompose so(4,1), for example at the reference point (2.6), into so(4) generators Mμν and translation generators Pμ=1RMμ0. Then C2[so(4,1)]=−R2PμPμ+C2[so(4)], and R2PμPμ∼−r−2[Xμ,[Xμ,⋅]] if acting on functions ϕ(x), cf. (3.69). Therefore S2∼C2[so(4)] should vanish on scalar functions ϕ(X) on H4, but not on higher-spin functions involving θab. We will see that this is indeed the case, and End(Hn) contains modes up to spin n as measured by S2 (3.55). We also observe(3.45)C2[so(4,2)]=C2[so(4,1)]−r−2□=S2−2r−2□,C2[so(4,2)]=2C2[so(4,1)]−S2. Note that S2,□, and C2[so(4,2)] commute and can be diagonalized simultaneously.Higher-spin modes on Hn4. To determine the spectrum of S2 for the modes in (3.42) we first prove the following identity for any f∈C:(3.46)S2({f,Xa}+)={S2f,Xa}+, where {⋅,⋅}+ denote the anti-commutator. To see this, consider(3.47)S2(fXa)=(S2f)Xa+[Mcd,f][Mcd,Xa]+2r−2[Xc,f][Xc,Xa]=(S2f)Xa+2i[Mad,f]Xd−2i[Xc,f]Mca=(S2f)Xa+2i[MadXd,f]−2iMad[Xd,f]−2i[Xc,f]Mca. Similarly,(3.48)S2(Xaf)=Xa(S2f)+2i[XdMad,f]−2i[Xd,f]Mad−2iMca[Xc,f] and adding them yields (3.46). Next, starting from(3.49)□XXa=−4Xa=−C2[so(5)]Xa this identity immediately implies(3.50)S2Pn(X)=0 for totally symmetrized polynomials Pn(X) in X. More generally, we show in appendix C.1 that this holds for any scalar field ϕ on H4 quantized via coherent states, i.e.(3.51)S2ϕ=0for anyϕ=∫CP1,2ϕ(x)|x〉〈x|. As a next step, we consider the higher-spin fields. Using(3.52)2C2[so(4,1)]Mab=[Mcd,[Mcd,Mab]]=12Mab□XMab=[Xc,[Xc,Mab]]=−2MabS2Mab=4Mab we find(3.53)S2ϕ(1)=4ϕ(1)for anyϕ(1)=ϕab(x)Mab with quantized functions ϕab(X) on H4 in (3.51), etc. We can similarly compute S2 for any irreducible polynomial function in Mab, and obtain(3.54)S2(Ξα_s)=2s(s+1)Ξsα_,Ξα_s=(Pα_)a1b1…asbsMa1b1…Masbs where Image 3 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(3.55) where Cs is the eigenspace of S2=2s(s+1). We refer to appendix C.1 for the details. Of course the Cs contain also forms of the type ϕα_(X)Ξα_s. However since the multiplication does not respect the grading, we can only say that Cs is the quantization of tensor fields ϕα_(x) taking values in the vector space spanned by Ξα_,(3.56)Cs∋Q(ϕa1…as;b1…bs(x)ma1b1…masbs)≡Q(ϕα_(x)Ξα_),s≤n, where Ξα_ denotes both the polynomials in Mab and mab. We remind the reader that Q (3.32) respects so(4,2). The truncation99We do not claim that for example the algebra of functions generated by Pa=Ma4 is truncated at order n; this is not the case. The claim is that all Hilbert-Schmidt operators can be written in the above way. 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 Sn2. See appendix C.1 for more details.This is a very remarkable structure, which leads to higher-spin fields on H4 truncated at spin n. For small n, the uncertainty scale LNC2≈R2, see (3.59), is set by the curvature scale of H4⊂R1,4, so that the space is far from classical. Nevertheless, the case of small n may be interesting after projection to the cosmological space-time M3,1 as discussed in section 6.The n=2 case. The case Nˆ=0=n−2 is special, because then C2[so(4,1)]=0=R2. To avoid this we will assume n≠2 in this paper.The n=0 case. In that case, (3.40) gives(3.57)ϵabcdefMabMcd=0, which is a relation in the Joseph ideal [28]. Then the Mab, 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 AdS5, but reduce Hn for n≠0 to the so(4,1) generators Mab, a,b=0,…,4, and the remaining Xa generators. Then the Mab, a,b=0,…,4 satisfy relations which are locally similar to the hs algebra of so(4,1), while the Xa generate the underlying space.3.6Semi-classical limit and Poisson calculusNow consider the semi-classical limit of fuzzy Hn4, which is obtained for large n, and is indicated by ∼. Then Xa∼xa and Θab∼θab, and the above relations on Hn4 reduce to(3.58a)xaxa=−R2,(3.58b)θabxb=0,(3.58c)ϵabcdeθabxc=nrθde∼2Rθde,(3.58d)γbb′≔ηaa′θabθa′b′=LNC44Pbb′, where the scale of non-commutativity is(3.59)LNC4≔θabθab=4r2R2. Here(3.60)Pab=ηab+1R2xaxbwithPabxb=0andPabPbc=Pac is the Euclidean projector on H4 (recall that H4 is a Euclidean space). Hence the algebra of functions on fuzzy Hn4 reduces for large n to the algebra of functions(3.61)End(Hn)∼C(CP1,2)=⊕sCs on the classical Poisson manifold CP1,2, as described in section 2.1. We denote the eigenspaces of S2 again with Cs, which are now modules over the algebra C0=C(H4) of functions on H4, thus encoding the structure of a bundle over H4. From now on we work in the semi-classical limit. The bundle structure can be made more explicit by writing(3.62)θab=ηiabJi as in (2.25), where ηiab are the tangential self-dual t'Hooft symbols; “tangential” follows from xaθab=0. The Ji transform as vectors of the local SU(2)L⊂SO(4), and describe the internal S2 fiber.Derivatives. It is useful to define the following derivations (cf. [2])(3.63)ðaϕ≔−1r2R2θab{xb,ϕ}=1r2R2xb{θab,ϕ},ϕ∈C, which are tangential xaða=0, satisfy the Leibniz rule, and are SO(4,1)-covariant. Equivalently,(3.64) In particular, the following holds:(3.65)ðaxc=−1r2R2θab{xb,xc}=PTac[ða,ðb]ϕ=−1r2R2{θab,ϕ} as shown in appendix D. The first line shows that ð act as isometries on functions, such that the ða can be viewed as a set of five Killing vector fields on H4 with Lie bracket given by (3.65). Furthermore,(3.66)ðaθcd=1r2R2θab{xb,θcd}=−1R2θab(ηbcxd−ηbdxc)=1R2(−θacxd+θadxc). We also note that the SO(4,1) rotations of scalar functions are generated by {Mab,⋅}, which can be written as(3.67){Mab,ϕ}=−(xaðb−xbða)ϕ,ϕ∈C0. To see this, it suffices to verify the action on the xc generators,(3.68){Mab,xc}=−(xaðb−xbða)xc=−(xaPbc−xbPac)=−(xaηbc−xbηac) 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:(3.69)□ϕ=−{xa,{xa,ϕ}}=−{xa,θabðbϕ}=−{xa,θab}ðbϕ−θab{xa,ðbϕ}=r2{Mab,xa}ðbϕ−θabθacðb∂cϕ=−r2R2Pabðaðbϕ for any ϕ∈C.Connection. We define an SO(4,1)-covariant connection on the module C (4.1) by [2](3.70)∇=PT∘ð so that for ∇a≡∇ða(3.71)∇aϕb=∂aϕb−1R2xbϕa,∇aϕbc=∂aϕbc−1R2(xbϕac+xcϕba) etc. if ϕa, ϕab are tangential. Comparing with (3.66) and using (3.67) it follows that the connection is compatible with θab, i.e.(3.72)∇θab=0,∇{f,g}={∇f,g}+{f,∇g} and ∇aPbc=0. The associated curvature(3.73)Rab≔R[ða,ðb]=[∇a,∇b]−∇[ða,ðb] is computed in appendix D.15, and reduces to the Levi–Civita connection on tensor fields. Thus Hn4 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 exists a canonical map(3.74)[⋅]0:C(CP1,2)→C(H4)f(ξ)↦f(x)=∫S2f(ξ) defined by integrating over the fiber at each x∈H4. This projects the functions on the total space to functions on the base space. On fuzzy SN4, this averaging can be defined in terms of a SO(5)-invariant projection to some sub-space of End(H). For Hn4, [⋅]0 is nothing but the projection to S2=0 i.e. to C0, as discussed below.Explicitly, the averaging [⋅]0 over the internal S2 is given by(3.75)[θabθcd]0=112LNC4(PacPbd−PbcPad+εabcde1Rxe)=r2R23(PacPbd−PbcPad+εabcde1Rxe). One can generalize the averaging to higher powers of θab, e.g. [2](3.76)[θabθcdθefθgh]0=35([θabθcd]0[θefθgh]0+[θabθef]0[θcdθgh]0+[θabθgh]0[θcdθef]0). Alternatively, one could proceed to define a star product for functions on H4, which is presumably commutative, but not associative, in analogy to the case of SN4 [11]. On the other hand, for n=0 there is nothing to project, and the full algebra of functions on H4 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,(3.77)TrQ(ϕ)=∫dμϕ=∫H4ρ[ϕ]0,ρ=ˆdim(H)Vol(H4) replacing the ill-defined fraction dim(H)Vol(H4) with the symplectic volume form dμ, which reduces to ρ on H4. 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 ða are not self-adjoint under the integral, but(3.78)∫ðafg=−∫fðag+1θR2∫f{xb,θab}g=−∫fðag−4R2∫xafg using {xb,θba}=4r2xa.4Functions, tensors and higher-spin modesWe have seen that the algebra End(Hn) of fuzzy Hn4 reduces in the semi-classical limit to the algebra of functions on CP1,2. The results of section 3.5 provide a more detailed decomposition of C into modules (3.55)(4.1) over the algebra of functions C0 on H4, due to (3.46). This means that C is a bundle over H4, whose structure is determined by the constraints (2.17), (2.21) and (2.24). An explicit description is given by the one-to-one map1010Note that Γ(s)H4 is not a module over C0, hence this is not a module isomorphism. In [2], a different convention was used for the map ϕa1…as(x)↔ϕ(s). The present convention avoids the appearance of square-roots of Casimirs in this map.(4.2) Here Γ(s)H4 denotes the space of totally symmetric, traceless, divergence-free rank s tensor fields on H4, which are identified with (symmetric tangential divergence-free traceless) tensor fields ϕa1…as(s) 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(4.3)Cs∋ϕ(s)↦{xa1,…{xas,ϕ(s)}…}0∈Γ(s)H4 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 Cs. Then the so(4,2) representation theory allows to characterize all polynomials in Cs 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](4.4)Cs∋ϕa1…as;b1…bs(s)(x)ma1b1…masbs↦ϕ(s)a1…as;b1…bs(x)xb1…xbs∈Γ(s)H4, which is equivalent to (4.3) up to normalization.Hence Cs encodes one and only one irreducible spin s field on H4, given by square-integrable tensor fields on H4. The generators Ξβ_ form a basis of irreducible totally symmetric polynomials in mab, i.e. of Young tableaux(4.5) As in [2], hs is closely related to the higher-spin algebra of Vasiliev theory.1111Note that H0 is a minirep of SO(4,2) but not of SO(4,1). This explains why we get an extension of Vasiliev's hs algebra by functions of X. Hence C can be viewed as functions on H4 taking values in hs.4.0.1Spin 1 modesThe unique spin 1 field is encoded in ϕabmab. According to the above statements, it can be expressed in terms of a tangential, divergence-free tensor field ϕa∈C0 on H4, i.e.(4.6)xaϕa=0=ðaϕa. Given such a ϕa, we define(4.7)ϕ(1)≔{xa,ϕa}=θabðbϕa=12θabFab∈C1,Fab=ðbϕa−ðaϕb which encodes the field strength of the vector field. This is not tangential, but(4.8)xaFab=xaðbϕa−xaðaϕb=xaðbϕa=−ϕb using (D.1). Conversely, the “potential” ϕa(x) is recovered from ϕ(1) via a projection(4.9) where ϕ(1)={xc,ϕc} for a tangential, divergence-free ϕa∈C0. The derivation of (4.9) is detailed in (D.18)–(D.20). The generalization of this formula for higher-spin is discussed below. If ϕa is an irrep of SO(4,1), we may abbreviate this as(4.10)−{xa,ϕ(1)}0=:αˆ1ϕa where αˆ1 is the value of α1(□−2r2) on ϕa.Pure gauge modes. Finally, one can verify that for ϕ˜a=ðaϕ, the associated “field strength” tensor is Fab∝{θab,ϕ}, but the field strength form ϕ(1) vanishes identically:(4.11)ϕ(1)={xa,ðaϕ}=0 using (D.3). This expresses the gauge invariance (or irreducibility) of ϕ(1).4.0.2Spin 2 modesSimilarly, spin 2 modes can be realized in terms of a tangential, divergence-free, traceless, symmetric rank 2 tensor ϕab(x)=ϕba(x)∈C0, i.e.(4.12)xaϕab=0=ðaϕab=ηabϕab. We define the associated “potential form”(4.13)ϕa(2)={xb,ϕab}=θbcðcϕab=−ωa;cbθcb∈C1, which can be viewed as so(4,1)-valued one-form with(4.14)ωa;cb=12(ðcϕab−ðbϕac). Note that ϕc(2) is indeed tangential,(4.15)xcϕc(2)=xc{xa,ϕca}=−{xa,xc}ϕca=0. The so(4)-valued components of ϕa(2) correspond to the spin connection, while its translational components(4.16)xcωa;cb=−12xcðbϕac=12ϕab reduce to ϕab, as on fuzzy SN4 [2]. The “field strength form” corresponding to ϕa(2) is(4.17)ϕ(2)={xa,ϕa(2)}=:12θadRad[ϕ]=−θadðd(ωa;cbθcb)=−θcbθadðdωa;cb−θadωa;cbðdθcb=12θadθcb(ðaωd;cb−ðdωa;cb)=:12θadθbcRad;bc[ϕ]∈C2 noting that the ðdθcb terms drop out for traceless, tangential ϕab, using (3.66). This encodes the linearized Riemann curvature tensor associated to ϕab,(4.18)Rad[ϕ]≔−ðaϕd(2)+ðdϕa(2)=Rad;bcθbc∈C1,Rad;bc=ðaωd;bc−ðdωa;bc=12(ðdðcϕab−ðaðcϕdb−ðdðbϕac+ðaðbϕdc). Although the ðe do not commute among another, their commutator is radial due to (3.67), i.e.(4.19)PRad;bc−PRbc;ad=0. Hence the tangential components of PRae;bc[ϕ] coincide with the usual linearized Riemann tensor. The connection form ϕc(2) (i.e. ω) is recovered by a projection(4.20) generalizing (4.9). Here, we defined ϕ(2)={xb,{xc,ϕbc}} for a tangential, divergence-free, traceless ϕab∈C0. 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 ϕab is an irrep of SO(4,1), we may abbreviate this as(4.21)−{xa,ϕ(2)}0=:αˆ2ϕa(2) where αˆ2 is the value of α2(□−2r2) on ϕa(2).Spin 2 pure gauge modes. Again, consider a pure gauge rank 2 tensor(4.22)ϕ˜ab(1)=∇aϕb+∇bϕa which is tangential and traceless (provided ðaϕa=0), but no longer divergence-free. Then(4.23)ϕ˜a(1)≔{xb,ϕ˜ab(1)}={xb,ðaϕb+ðbϕa−1R2(xaϕb+xbϕa)}={xb,ðaϕb}−1R2{xb,xaϕb}=ðaϕ(1)+2R2θacϕc using (D.7) and ϕ(1)={xa,ϕa}. This satisfies(4.24){xa,ϕ˜a(1)}={xa,ðaϕ(1)+2R2θacϕc}=2R2{xa,θacϕc}=0 using (5.30), which expresses the gauge invariance of ϕ(2).4.0.3Spin s modes and Young diagramsAs observed above, elements in Cs can be identified with totally symmetric, traceless, divergence-free rank s tensor fields ϕa1…as on H4 via(4.25)ϕ(s)={xa1,…{xas,ϕa1…as}…}∈Cs. It is useful to define also the mixed spin s objects, such as the “connection (2s−1)-form”(4.26)ϕa(s)={xa1,…,{xas−1,ϕa1…as−1a}…}∈Cs−1 which are all tangential and associated to the underlying irreducible rank s tensor field. Then the “field strength” form can be written as(4.27)ϕ(s)={xa,ϕa(s)}=:12Rad[ϕ]θad=θa1b1ðb1…θasbsðbsϕa1…as=θa1b1…θasbsðb1…ðbsϕa1…as=:Ra1…as;b1…bs(x)θa1b1…θasbs≡Rα_(x)Ξα_∈Cs noting that the ðθ… terms drop out for traceless tangential ϕa1…as, using (3.66). Here(4.28)Rad[ϕ]≔−ðaϕd(s)+ðdϕa(s)Rb1…bs;a1…as(x)=Pða1…ðasϕb1…bs is some antisymmetrized derivatives corresponding to some two row rectangular Young projector Image 11, which can be regarded as linearized higher-spin curvature. We will show below that the potential ϕa(s) is then recovered from the following projection(4.29) If the underlying ϕa1…as∈C0 is an irrep of SO(4,1), we may abbreviate this as(4.30)−{xa,ϕ(s)}s−1=:αˆsϕa(s) where αˆs is the value of αs(□−2r2) on ϕa(s).Pure gauge modes. Finally, one can verify that a pure gauge rank s tensor(4.31)ϕ˜a1…as(s−1)=∇(asϕa1…as−1) drops out from the field strength form,(4.32){xa1,…,{xas,ϕ˜a1…as(s−1)}…}=0. 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(4.33)ϕ(s)↦{Λ,ϕ(s)}. These correspond to symplectomorphisms on CP1,2 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 5.9.4.1Inner product and quadratic actionIt 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(4.34)∫ϕ(1)ϕ(1)=14∫[θabθcd]0FabFcd=r2R212∫(2PacPbd+xfRεabcdf)FabFcd which looks like the action for self-dual (abelian) Yang–Mills. In the spin 2 case, consider the analogous quadratic form(4.35)∫ϕ(2)ϕ(2)=14∫[θaeθbcθa′e′θb′c′]0Rae;bc[ϕ]Ra′e′;b′c′[ϕ]=310∫[θaeθa′e′]0[θbcθb′c′]0Rae;bc[ϕ]Ra′e′;b′c′[ϕ]=130∫(2Paa′Pee′+xfRεaea′e′f)(2Pbb′Pcc′+xfRεbcb′c′f)Rae;bc[ϕ]Ra′e′;b′c′[ϕ]=215∫Paa′Pee′Pbb′Pcc′Rae;bc[ϕ]Ra′e′;b′c′[ϕ]+topological terms because [ϕ(2)]0=0. Note that we used the symmetries (4.19) of Rae;bc or rather of its tangential part PRae;bc, as the radial contributions drop out anyway. We observe that (4.35) is a (self-dual) linearized quadratic gravity action,1212The topological terms are the linearized Pontryagin and Euler class (i.e. Gauss–Bonnet term). which can be written in terms of the Rab “forms” as follows:(4.36)∫ϕ(2)ϕ(2)=14∫[θaeθbcθa′e′θb′c′]0Rae;bcRa′e′;b′c′=310∫[θaeθa′e′]0Rae;bcθbcRa′e′;b′c′θb′c′=310∫[θaeθa′e′]0RaeRa′e′=2r2R25∫(PacPbd−PadPbc+xeRεabcde)ðbϕa(2)ðdϕc(2). Similarly for spin s, we have(4.37)∫ϕ(s)ϕ(s)=∫[θaeθa′e′RaeRa′e′]0=αsr2R2∫(PacPbd−PadPbc+xeRεabcde)ðbϕa(s)ðdϕc(s) which is again some self-dual quadratic Fronsdal-type higher-spin action [29]. The factor αs 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 [30]. In the remainder of this paper, we will show how a non-trivial higher-spin gauge theory arises from multi-matrix models.4.1.1Projections, positivity and determination of αsNow consider the spin s modes ϕ(s)∈Cs as above, determined by some irreducible rank s tensor field on H4. We have seen that this in one-to-one correspondence to a spin s potential ϕa(s)∈Cs−1 as above. Then(4.38)−∫ϕa(s){xa,ϕ(s)}s−1=−∫ϕa(s){xa,ϕ(s)}=∫{xa,ϕa(s)}ϕ(s)=∫ϕ(s)ϕ(s). This provides the following relations:Spin 1 case. For spin s=1, the projection {xa,ϕ(s)}0 in (4.38) was computed in (4.9), which gives(4.39)∫ϕ(1)ϕ(1)=α1∫ϕa(1)(□−2r2)ϕa(1)≥0, for Hermitian ϕ(1) Therefore(4.40)α1=13, and in particular □−2r2 is positive on C0.Spin 2 case. We can evaluate the right-hand side of (4.38) using (4.36) as(4.41)∫ϕ(2)ϕ(2)=2r2R25∫Pacðbϕa(2)ðbϕc(2)−ðdϕa(2)ðaϕd(2)+xeRεabcdeðbϕa(2)ðdϕc(2)=2r2R25∫−ϕa(2)ðbðbϕa(2)+1R2ϕb(2)ϕb(2)−4R2ϕa(2)ϕa(2)+1r2R2ϕa(2)({θad,ϕd(2)}−12Rεabdcexe{θbd,ϕc(2)})=α2∫ϕa(2)(□−2r2)ϕa(2) using (3.78), (D.22), the self-duality relation (D.23) and (D.24). Therefore(4.42)α2=25. This holds in fact for any tangential divergence-free ϕc∈C1. Together with (4.38), this establishes the formula (4.20). On the other hand, (4.40) and (4.39) implies also e.g.1313Since ϕa(2)={xc,ϕcb} for tangential traceless divergence-free ϕab∈C0; the index a is irrelevant here.(4.43)∫ϕa(2)ϕa(2)=13∫ϕab(□−2r2)ϕab if ϕab∈C0 is divergence-free, traceless and tangential (by fixing one index).Generic spin s case. In the generic case, we obtain similarly(4.44)∫ϕ(s)ϕ(s)=∫ϕa(s)αˆsϕa(s),αˆsϕ(s)=αs(□−2r2)ϕa(s),(4.45)∫ϕ(s)ϕ(s)=∫ϕa1…as(s)αˆs…αˆ2αˆ1ϕa1…as(s). Explicit expressions for αs for s≥3 could be computed similarly but are not required for our purposes.4.2Local decompositionFinally consider any point on H4, for instance the reference point (2.6). We denote the four tangential coordinates with xμ, and the time-like coordinate on R4,1 with x0. Then the so(4,1) generators decompose (locally) into so(4) generators mμν, and the remaining translation generator by pμ=mμ0. We can then decompose e.g. the spin s=1 modes locally as(4.46)ϕab(x)mab=ϕμ(x)pμ+ϕμν(x)mμν∈C1 and similar for higher-spin. From this point of view, the main lesson of the above results is that the ϕμ(x) and ϕμν(x) are not independent fields, but determined by the same irreducible spin 1 field ϕa(x), and similarly for higher-spin fields. For generalized fuzzy spaces these constraints may disappear, as considered in [3]. For the basic spaces Hn4 and for SN4 [2], the formalism developed above takes these constraints properly into account.5Matrix model realization and fluctuationsNow consider the IKKT matrix models with mass term,(5.1)S[Y]=1g2Tr([Ya,Yb][Ya′,Yb′]ηaa′ηbb′−μ2YaYa). Here ηab=diag(−1,1,…,1) is interpreted as Minkowski metric of the target space R1,D−1. The positive mass μ2>0 should ensure stability. The above model leads to the classical equations of motion(5.2)□YYa+12μ2Ya=0 where(5.3)□Y=[Ya,[Ya,⋅]]∼−{ya,{ya,⋅}} plays the role of the Laplacian. Note that (5.2) are precisely the equation of motions for the IKKT model put forward in [31] after taking an IR cutoff into account.5.1Fuzzy Hn4 solution and tangential fluctuation modesConsider the solution Ya=Xa of (3.6) corresponding to fuzzy Hn4, and add fluctuations(5.4)Ya=Xa+Aa on Hn4. They naturally separate into tangential modes xaAa=0 and radial modes xaAa≠0. The SO(4,1)-invariant inner product(5.5)〈A(i),A(j)〉≔∫Aa(i)Ab(j)ηab is positive definite for (Hermitian) tangential Aa on H4, and negative for the radial modes. Since Aa∈End(Hn)⊗R5, we expect four tangential fluctuation modes and one radial mode for each spin (except for spin 0), as for SN4 [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(5.6)I(Aa)≔{θab,Ab}I˜(Aa)≔Paa′{θa′b,Ab}G(Aa)≔{xa,{xb,Ab}}. They are Hermitian w.r.t. the inner product (5.5), and tangential except for I, noting that(5.7)xaI(Aa)=xa{θac,Ac}=−r2R2ðaAaðaI(Aa)=ða{θad,Ad}=1r2R2xb{θab,{θad,Ad}}=−1r2R2xbI2(Ab). The SO(4,1) Casimir for the fluctuation modes can be expressed using I as follows:(5.8)C2[so(4,1)](full)Aa=12([Mcd,⋅]+Mcd(5))2Aa=C2[so(4,1)](ad)Aa−2r−2I(Aa)+4=(−r−2□−2r−2I+S2+4)Aa=(R2ð⋅ð−2r−2I+S2+4)Aa using (3.44), and C2[so(4,1)]=4 for the vector representation C5. This can be seen by expressing I as follows:(5.9)−θ(Mcd(ad)⊗Mcd(5)A)a∼−(Mcd(5))bai{θcd,⋅}Ab=2{θab,Ab}=2I(A)a. Here(5.10)(Mab(5))dc=i(δbcηad−δacηbd) is the vector generator of so(4,1), and Mbc(ad)=i{Mbc,⋅} denotes the representation of so(4,1) induced by the Poisson structure on S4. As a check, we note that C(full)(xa)=0, since I(xa)=4xa. This reflects the full SO(4,1)-invariance of the background xa.5.1.1Spin 0 modesLet ϕ∈C0 be a spin 0 scalar field. There are two tangential spin 0 modes, which read(5.11)Aa(1)=ðaϕ∈C0,ϕ∈C0,Aa(2)=θabðbϕ={xa,ϕ}∈C1. These modes satisfy(5.12){xa,Aa(1)}={xa,ðaϕ}=0,{xa,Aa(2)}=−□ϕ, using (D.3). Clearly only Aa(1) is physical, while Aa(2) is a pure gauge field. Let us compute the action of the I intertwiner; to start with(5.13)I(Aa(2))≔{θab,{xb,ϕ}}={{θab,xb},ϕ}}+{xb,{θab,ϕ}}=4r2{xa,ϕ}+r2{xb,(xaðb−xbða)ϕ}=4r2{xa,ϕ}+r2θbaðbϕ=3r2A(2). Similarly, one finds(5.14)I(Aa(1))≔{θab,ðbϕ}=r2Aa(1) Now we can use the identities(5.15)I(Aa(2))={θab,θbb′Ab′(1)}={θab,θbb′}Ab′(1)+θbb′{θab,Ab′(1)}=3r2θab′Ab′(1)+{θbb′θab,Ab′(1)}−θab{θbb′,Ab′(1)}=3r2θab′Ab′(1)−r2R2{Pab′,Ab′(1)}−θabI(Ab(1))=3r2θab′Ab′(1)−r2(xa{xc,Ac(1)}−θacAc(1))−θabI(Ab(1))=4r2θacAc(1)−θabI(Ab(1)), wherein we used(5.16)R2{Pab,ϕb}={xaxc,ϕc}=xa{xc,ϕc}+xc{xa,ϕc}=xa{xc,ϕc}−θacϕc, for any tangential ϕc, and the gauge fixing relations (5.30). Therefore(5.17)θabI(Ab(1))=4r2Aa(2)−I(Aa(2))I˜(Aa(1))=4r2Aa(1)+1r2R2θadI(Ad(2)). For s=0, this gives(5.18)I(Aa(2))=3r2A(2) since S2A(2)=4A(2), in agreement with (5.13). Then (5.17) gives(5.19)I˜(Aa(1))=r2Aa(1) because Aa(1) is tangential. To summarize,(5.20)I˜(Aa(1)Aa(2))=r2(1003)(Aa(1)Aa(2))5.1.2Spin 1 modesNow let ϕa∈C0 be a tangential, divergence-free spin 1 field. Then there are four tangential spin 1 modes, given by(5.21)Aa(1)=ðaϕ(1)∈C1,ϕ(1)={xa,ϕa}∈C1Aa(2)=θabðbϕ(1)={xa,ϕ(1)}∈C2⊕C0,Aa(3)=ϕa∈C0,Aa(4)=θabϕb∈C1. Here ϕ(1) is the unique spin 1 mode in End(H). I can be computed on the Aa(3) and Aa(4) modes using(5.22)I(ϕa)≔{θab,ϕb}=r2ϕa due to (D.2), which gives(5.23)I(Aa(3))≔{θab,ϕb}=r2ϕa=r2Aa(3). Furthermore,(5.24)I˜(Aa(4))≔Paa′{θa′b,θbcϕc}=Paa′θbc{θa′b,ϕc}+Paa′{θa′b,θbc}ϕc=Paa′{θbcθa′b,ϕc}−θab{θbc,ϕc}+3r2θacϕc=−r2R2Paa′{Pa′c,ϕc}+2r2θacϕc=r2Paa′θa′cϕc+2r2A(4)a=3r2Aa(4) using (5.16). I(Aa(2)) and I(Aa(1)) will be computed for the general case below.5.1.3Spin 2 modesNow let ϕab=ϕba∈C0 be a tangential, divergence-free, traceless spin 2 field, and let ϕa(2)={xb,ϕab}∈C1. Then there are four tangential spin 2 modes, given by(5.25)Aa(1)=ðaϕ(2)∈C2,ϕ(2)={xa,ϕa(2)}∈C2,Aa(2)=θabðbϕ(2)={xa,ϕ(2)}∈C3⊕C1,Aa(3)=ϕa(2)∈C1,Aa(4)=θabϕb(2)∈C2. 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 ϕc(2) is indeed tangential. Furthermore,(5.26)I(Aa(3))≔{θab,ϕb(2)}={θab,{xc,ϕbc}}=−{ϕbc,{θab,xc}}−{xc,{ϕbc,θab}}=r2{ϕbc,ηacxb−ηbcxa}+r2{xc,ϕac}=0 using (5.22) for the last term. Adapting (5.27), we obtain(5.27)I˜(Aa(4))=Paa′{θbcθa′b,ϕc}−θab{θbc,ϕc}+3r2θacϕc=−r2R2Paa′{Pa′c,ϕc}+3r2θacϕc=r2Paa′θa′cϕc+3r2A(4)=4r2A(4). It is illuminating to display the explicit tensor content of the spin 2 modes, recalling that ϕb(2) is the spin connection (4.13) and ϕ(2) is the curvature tensor. Using (4.17), this is(5.28)Aa(1)=ðaϕ(2)=12ða(θedθbcRed;bc[ϕ(2)]),Aa(2)=12θaa′ða′(θedθbcRed;bc[ϕ(2)]),Aa(3)=−ωa;deθde,Aa(4)=−θabωb;deθde. In particular, Aa(4)=θabAb encodes a so(4,1)-valued gauge field Ab=−ωb;deθde given by the linearized spin connection of ϕab.5.1.4Spin s≥1 modesNow consider the generic case. Let ϕa1…as∈C0 be a tangential, divergence-free, traceless, symmetric spin s field, and let ϕa(s)={xa1,…{xas−1,ϕa1…as−1a}…}∈Cs−1. Then there are four tangential spin s modes, given by(5.29) Here ϕ(s) is the unique spin s mode in End(H). The modes (5.29) satisfy the gauge-fixing relations(5.30){xa,Aa(1)}={xa,ðaϕ(1)}=0,{xa,Aa(4)}={xa,θabϕb}=θab{xa,ϕb}=r2R2ðaϕa=0,{xa,Aa(3)}={xa,ϕa}=ϕ(1),{xa,Aa(2)}=−□ϕ(1), using (D.3), and(5.31)ðaAa(1)=−1r2R2□ϕ(1),ðaAa(2)=0=ðaAa(3),ðaAa(4)=ða(θabϕb)={xa,ϕa}=ϕ(1). 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 to I˜; we first show that(5.32)I˜(Aa(3))≔{θab,ϕb(s)}=(2−s)r2Aa(3) This can be proven inductively as follows:(5.33){θab,ϕb(s)}={θab,{xc,ϕbc(s)}}=−{ϕbc(s),{θab,xc}}−{xc,{ϕbc(s),θab}}=r2{ϕba(s),xb}+(3−s)r2{xc,ϕac(s)}=(2−s)r2ϕa(s) using (5.22), where ϕab(s)={xa1,…,{xas−2,ϕa1…as−2ab}…}∈Cs−2. Note that we employed the relation {θab,ϕbc(s)}=(3−s)r2ϕbc(s) for ϕbc(s)∈Cs−2, which can be derived via induction, too. Adapting (5.27), this yields(5.34)I˜(Aa(4))=Paa′{θbcθa′b,ϕc}−θab{θbc,ϕc}+3r2θacϕc=−r2R2Paa′{Pa′c,ϕc}−(2−s)r2θabϕb(s)+3r2θacϕc=r2Paa′θa′cϕc+(s+1)ϕa(s)r2A(4)=(s+2)r2A(4). To compute I(Aa(2)), consider(5.35)I(Aa(2))≔{θab,Ab(2)}={{xa,xb},Ab(2)}=−{{xb,Ab(2)},xa}−{{Ab(2),xa},xb}, where the second term can be rewritten as(5.36)−{{Ab(2),xa},xb}=−{{{xb,ϕ},xa},xb}={{{ϕ,xa},xb},xb}+{{θab,ϕ},xb}=□Aa(2)−{{ϕ,xb},θab}−{{xb,θab},ϕ}=□Aa(2)−I(Ab(2))+4r2Aa(2). So that we obtain(5.37)2I(Aa(2))=−{{xb,Ab(2)},xa}+□Aa(2)+4r2Aa(2)={□ϕ,xa}+(□+4r2)Aa(2) for any spin s≥1. Therefore(5.38)(□−2I)(Aa(2))=−{□ϕ,xa}−4r2Aa(2). On the other hand, for a spin s field ϕ we have(5.39){xa,□ϕ}=A(2)[□ϕ]=r2A(2)[(−C2+S2)ϕ]=r2(2s(s+1)−Cfull2)A(2)[ϕ]=((□+2I)−r2S2+2r2s(s+1)−4r2)A(2)[ϕ], using the intertwiner property and (5.8), hence(5.40){xa,□ϕ}−□{xa,ϕ}=(2I−r2S2+2r2s(s+1)−4r2)A(2)[ϕ]. Comparing with (5.37), this gives2I(Aa(2))=−{xa,□ϕ}+(□+4r2)Aa(2)=(4r2−2I+r2S2−2r2s(s+1)+4r2)A(2) such that(5.41) for s≥1, which is tangential. Hence if S2 is diagonal then I is also diagonal, and the Casimir C2[SO(4,1)] (5.8) can be diagonalized simultaneously. To evaluate (5.41), we decompose Aa(2) into its components in Cs−1⊕Cs+1 as follows:(5.42)Aa(2)=−αs(□−2r2)Aa(3)+Aa(2′)∈Cs−1⊕Cs+1,Aa(2′)≔Aa(2)+αs(□−2r2)Aa(3)∈Cs+1, using (4.29); recall that Aa(3)≡ϕa(s). Note that (5.42) is simultaneously a decomposition into eigenvectors of I,(5.43)2I(Aa(2′))=2(s+3)r2Aa(2′),2I(Aa(3))=2(2−s)r2Aa(3) consistent with (5.32). Then we arrive at(5.44)2I(Aa(2))=−2(2−s)αsr2(□−2r2)Aa(3)+2(s+3)r2Aa(2′)=2(s+3)r2Aa(2)+2(2s+1)αsr2(□−2r2)Aa(3) and I(Aa(1)) is obtained from (5.17),(5.45)I˜(Aa(1))=4r2Aa(1)+1R2θad((s+3)Ad(2)+(2s+1)αs(□−2r2)Ad(3))=r2(1−s)Aa(1)+2s+1R2αsθab(□−2r2)Aa(3)=r2(1−s)Aa(1)+2s+1R2αˆsAa(4) where the last line is only a short-hand notation which applies to irreps, cf. (4.10). Hence I˜ is diagonalized as follows:(5.46)I˜(Aa(1′))=r2(1−s)Aa(1′),Aa(1′)=Aa(1)−αsR2r2θ˜ab(□−2r2)Ab(3)≡Aa(1)−αˆsR2r2Aa(4), using (5.34). Accordingly, we define the eigenmodes(5.47)(Ba(1),Ba(2),Ba(3),Ba(4))≔(Aa(1′),Aa(2′),Aa(3),Aa(4)), which satisfy(5.48a)I˜(Ba(1)Ba(2)Ba(3)Ba(4))=r2(1−s0000s+300002−s00002+s)(Ba(1)Ba(2)Ba(3)Ba(4)),(5.48b)S2(Ba(1)Ba(2)Ba(3)Ba(4))=2(s(s+1)0000(s+1)(s+2)0000(s−1)s0000s(s+1))(Ba(1)Ba(2)Ba(3)Ba(4)). 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(5.49)G(Aa(2))=−{xa,□ϕ(s)}G(Aa(3))={xa,ϕ(s)}=Aa(2)G(Aa(1))=G(Aa(4))=0.5.2Recombination, hs-valued gauge fields and Young diagramsThe 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 ϕa1…as are irreducible, so that the modes can be captured in a simpler way.Trace contributions. These arise from(5.50)ϕa1…as=ηa1a2ϕa3…as. Then(5.51)ϕ˜a(s)={xa1,…{xas−1,ηa1a2ϕa3…a}…}}=−□ϕ(s−2)a∈Cs−2,ϕ˜(s)={xa,ϕ˜a(s)}=−{xa,□ϕa(s)} which enters the four modes as follows(5.52)Aa(1)=ðaϕ˜(s−2)∈Cs−2,Aa(2)=θabðbϕ˜(s−2)={xa,ϕ(s−2)}∈Cs−1⊕Cs−3Aa(3)=ϕ˜a(s−2)∈Cs−3,Aa(4)=θabϕ˜b(s−2)∈Cs−2. Hence the trace components reproduce the four modes with spin s−2, as long as □ϕa(s)≠0.Divergence modes. Now we drop the requirement that ϕab is divergence-free. Consider the case of rank 2 tensors, expressed in terms of spin 1 modes as in (4.22)(5.53)ϕ˜ab(1)=∇aϕb+∇bϕa. Then according to (4.24), these contributions to the would-be spin 2 modes Aa(1),Aa(2) vanish identically. The contribution to Aa(3) reduces to a combinations of the spin 1 modes of Aa(1) and Aa(4), and the contribution to Aa(4) reduces to a combinations of the spin 1 modes of Aa(2) and Aa(3). Hence if we drop the divergence-free condition, it would suffice to keep the Aa(3) and Aa(4) modes.1414However, the spin 0 modes cannot be recovered from divergence modes: for ϕa=ðaϕ we get ϕ˜(1)={xa,ðaϕ}=0 due to (D.3). In particular, we need not worry about these constraints upon projecting H4 to M3,1. It will suffice to impose the appropriate divergence- and trace-conditions for M3,1.Finally as for SN4 [2], we can collect all tangential fluctuation modes as hs-valued tangential gauge fields(5.54)Aa=θacAc,Ac=Ac,α_(x)Ξα_ where Ac,α_(x) are double-traceless tensor fields corresponding to 2-row Young diagrams of the type Image 15. The external leg is associated to the extra box in the Young diagram. However the Ac,α_(x) are in fact higher curvatures of the underlying symmetric tensor fields ϕa1…as as in (4.28), which characterize the irreducible physical degrees of freedom Aa(i).5.2.1Inner productsThe inner products (5.5) of the tangential fluctuations are given by(5.55)∫Ab(1)Ab(1)=∫ðbϕ(s)ðbϕ(s)=αsr2R2∫ϕa(s)(□+2r2s)(□−2r2)ϕa(s),∫Ab(1)Ab(4)=∫ðaϕ(s)θabϕb(s)=∫ϕ(s){xb,ϕb(s)}=∫ϕ(s,1)ϕ(s,4)=αs∫ϕa(s)(□−2r2)ϕa(s),∫Ab(3)Ab(2)=∫ϕb(s){xb,ϕ(s)}=−∫{xb,ϕb(s)}ϕ(s)=−αs∫ϕa(s)(□−2r2)ϕ(s)a,∫Ab(2)Ab(2)=∫{xb,ϕ(s)}{xb,ϕ(s)}=αs∫ϕa(s)(□+2r2s)(□−2r2)ϕa(s),∫Aa(3)Aa(3)=∫ϕb(s)ϕb(s),∫Aa(4)Aa(4)=∫θabϕb(s)θacϕc(s)=r2R2∫ϕb(s)ϕb(s),∫Ab(1)Ab(2)=∫Ab(1)Ab(3)=∫Aa(4)Aa(2)=∫Aa(4)Aa(3)=0, using (3.69), (D.3), (D.36) and [θabϕb(s,4)ϕa(s,3)]0=0; we drop the labels ϕb(s,4)≡ϕb(s) if no confusion can arise.Now consider the eigenstates (5.47) of I. We verify that Ba(2) and Ba(1) satisfy the orthogonality relations(5.56)∫Ba(2)Ba(3)=∫(Aa(2)+αˆsAa(3))Aa(3)=0,∫Ba(1)Ba(4)=0, using the definitions (5.42), (5.46) as well as (4.44). Therefore {Ba(i)}i=14 form an orthogonal basis of eigenmodes. The normalization can be computed as(5.57)∫Ba(2)Ba(2)=∫(Aa(2)+αˆsAa(3))(Aa(2)+αˆsAa(3))=αs∫ϕa(2′)((□+2r2s)−αs(□−2r2))(□−2r2)ϕa(2′)=∫ϕ(2′)((1−αs)□+2r2(s+1))ϕ(2′) and(5.58)∫Ba(1)Ba(1)=∫(Aa(1)−αˆsR2r2Aa(4))(Aa(1)−αˆsR2r2Aa(4))=αsr2R2∫ϕa(1′)((□+2r2s)−αs(□−2r2))(□−2r2)ϕ(1′)a=1r2R2∫ϕ(1′)((1−αs)□+2r2αs(s+1))ϕ(1′). Note that all Aa(2) modes are pure gauge modes, and they will drop out in the action.5.3Radial modesFinally consider the radial fluctuation modes. These are given by(5.59)Aa(r)[ϕ(s)]=xaϕ(s),ϕ(s)∈Cs. They are dangerous because the radial metric in R1,4 is negative,(5.60)∫Ab(r)Ab(r)=∫xaϕ(s)xaϕ(s)=−R2∫ϕ(s)ϕ(s) recalling that xaxa=−R2<0. However, they disappear after the projection to M3,1. If we include these radial fluctuations, we should first diagonalize I. We have(5.61)I(Aa(r))={θab,xbϕ(s)}={θab,xb}ϕ(s)+xb{θab,ϕ(s)}=4xaϕ(s)−θab{xb,ϕ(s)}=4Aa(r)+r2R2Aa(1). Recall that I(A(2′,3)) is tangential, but I(A(1,4)) is not, with(5.62)xaI(A(1)[ϕ])=□ϕ,xaI(A(4)[ϕ])=−r2R2ϕ,xaI(A(1′)[ϕ])=(□+αˆs)ϕ, 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 M4,1 in the next stage.5.4SO(4,1)-invariant quadratic action on H4The quadratic fluctuations for the fluctuation modes ya=xa+Aa are governed by the action(5.63)S[y]=S[x]+S2[A]+O(A3), where(5.64)S2[A]=2g2∫dμ(Aa(D2A)a+{xa,Aa}2)=2g2∫dμAa(D2+G)Aa. Here(5.65)(D2A)≔(□−2I+12μ2)A is the “vector” (matrix) Laplacian, and G(A) (5.6) ensures gauge invariance. The mass term determines r2 via the on-shell condition for Hn4,(5.66)0=(□+12μ2)xa,12μ2=4r2.Gauge-invariant action. Consider first the gauge-invariant kinetic term(5.67)S2[A]=2g2∫dμAa(D2+G)Aa. We verify that the pure gauge modes Aa(2) are null modes using (5.49) and (5.38):(5.68)(D2+G)Aa(2)=−{□ϕ(s),xa}+(12μ2−4r2)Aa(2)−{xa,□ϕ(s)}=0 for any spin, taking into account the on-shell condition 12μ2=4r2. Hence the pure gauge modes Aa(2) indeed decouple.For spin 0, we determine the action explicitly for the B(1) and B(2) modes(5.69)(D2+G)(Ba(1)Ba(2))=(□+2r2000)(Ba(1)Ba(2)). The inner product is diagonal for spin 0, and the quadratic action is given by(5.70)S2[A]=∫Ba(i)(□+2r2000)Ba(i). Since Ba(1)∈C0, 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:(5.71)S2,(fix)[A]=2g2∫dμAD2A. We work in the basis {B(i)} (5.47) where I is diagonal. Then the eigenvalues of the kinetic operator D2 are elaborated in the appendix D.1. Together with the inner products in section 5.2.1, we obtain the following diagonalized quadratic action(5.72a)∫Ba(1)D2B(1)a=αsr2R2∫ϕa(s)×((□+2r2s)−αs(□−2r2))(□−2r2)(□+2r2(3s+2))ϕa(s)=1r2R2∫ϕ(s)((1−αs)□+2r2αs(s+1))(□+4r2(s+1))ϕ(s),(5.72b)∫Ba(2)D2B(2)a=αs∫ϕa(s)((□+2r2s)−αs(□−2r2))(□−2r2)(□+2r2s)ϕ(s)a=∫ϕ(s)((1−αs)□+2r2αs(s+1))□ϕ(s),(5.72c)∫Ba(3)D2Ba(3)=∫ϕb(s)(□+2r2s)ϕb(s),(5.72d)∫Ba(4)D2Ba(4)=r2R2∫ϕb(s)(□−2r2s)ϕb(s). All these terms are non-negative, because(5.73)(□+2r2s)−αs(□−2r2)=(1−αs)□+2r2(s+αs)>0.A[(□−2r2s)ϕb(s)]∝A[(□+r2s(s−3))ϕa1…as] for any intertwiner A, using (D.33). The first line is positive because 1>αs, and the second line is positive since □+r2s(s−3) is manifestly positive for s≥3, while for s=1,2 it coincides with □−2r2 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(5.74a)∫Ba(1)D2B(1)a=α1r2R2∫ϕa((□+2r2)−α1(□−2r2))(□−2r2)(□+10r2)ϕa=1r2R2∫ϕ(1)((1−α1)□+4r2α1)(□+8r2)ϕ(1),(5.74b)∫Ba(2)D2Ba(2)=α1∫ϕa((□+2r2)−α1(□−2r2))(□−2r2)(□+2r2)ϕa=∫ϕ(1)((1−α1)□+4r2α1)□ϕ(1),(5.74c)∫Ba(3)D2Ba(3)=∫ϕa(□+2r2)ϕa,(5.74d)∫Ba(4)D2Ba(4)=r2R2∫ϕa(□−2r2)ϕa, and for spin 2(5.75a)∫Ba(1)D2B(1)a=α2r2R2∫ϕa(2)((□+4r2)−α2(□−2r2))(□−2r2)(□+16r2)ϕa(2)=1r2R2∫ϕ(2)((1−α2)□+6r2α2)(□+12r2)ϕ(2)=α1α2r2R2∫ϕab(□+6r2−α2□)(□+18r2)(□−2r2)□ϕab,(5.75b)∫Ba(2)D2Ba(2)=α2∫ϕa(2)((□+4r2)−α2(□−2r2))(□−2r2)(□+4r2)ϕa(2)=∫ϕ(2)((1−α2)□+6r2α2)□ϕ(2)=α2α1∫ϕab(□+6r2−α2□)(□+6r2)(□−2r2)□ϕab,(5.75c)∫Ba(3)D2B(3)a=∫ϕa(2)(□+4r2)ϕa(2)=α1∫ϕab(□+6r2)(□−2r2)ϕab,(5.75d)∫Ba(4)D2Ba(4)=r2R2∫ϕb(2)(□−4r2)ϕb(2)=α1r2R2∫ϕab(□−2r2)2ϕab, 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 M3,1, as discussed below.5.5Yang–Mills gauge theoryWe can write the full action (5.1) in a conventional (higher-spin) Yang–Mills form for the recombined higher-spin gauge fields (5.54) Aa=θabAb. Then the field strength is(5.76)Fab=[Xa+Aa,Xb+Ab]∼θab+θaa′θbb′Fa′b′,Fab=∇aAb−∇bAa′+[Aa,Ab] recalling that ∇θab=0. Hence the action (5.1)(5.77)S[Y]∼1gYM2∫H4(FabFa′b′ηaa′ηbb′−2R2AaAa′ηaa′) is basically a hs-valued Yang–Mills action1515We used xaAa=0; the apparent “mass” term is at the cosmological curvature scale, and would presumably disappear upon imposing the non-linear constraint YaYa=−R2. (dropping surface terms and using μ2=8r2), where(5.78)1gYM2=ρLNC84g2 is the dimensionless Yang–Mills coupling constant. For nonabelian spin 1 modes Aa(4) on stacks of Hn4 branes, the usual Yang–Mills action is recovered. For spin 2, one would expect this to describe some type of quadratic gravity action [32–34]. 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.5.6Metric and gravitons on H4Now 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 H4. These contributions are expected to give rise to linearized gravity. The kinetic term for all fluctuations on a given background Ya∼ya arises in the matrix model from1616One might worry about the contributions from {ya,⋅} on the generators θbc for higher-spin modes. However the metric is always defined by the two derivative terms acting on the tensor fields.(5.79)S[ϕ]=−Tr[Ya,ϕ][Ya,ϕ]∼∫ρ{ya,ϕ}{ya,ϕ}=∫ργabðaϕðbϕ=ξ∫ργμν∂μϕ∂νϕ=∫H4d4x|Gμν|Gμν∂μφ∂νφ using (3.77); some dimensionful constants are absorbed in φ, and Greek indices indicate local coordinates. Here γab is a symmetric tensor in SO(4,1) notation(5.80)γab=ηcc′ecaec′b,eca={yc,xa} which in local coordinates near some reference point ξ reduces to γμν, cf. (3.58d). Hence the effective metric is given by [3,15,19](5.81)Gμν=4αLNC4γμν,α=LNC44|γμν| and eca can be interpreted as vielbein. For a deformation of the H4 background of the form(5.82)ya=xa+Aa, the metric is perturbed due to γab=γ‾ab+δAγab+O(A2) with(5.83)δAγab=:Hab[A]={xc,xa}{Ac,xb}+(a↔b)=θca{Ac,xb}+(a↔b)={θcaAc,xb}+{θcbAc,xa}+r2(Abxa+Aaxb−2ηab(Acxc)). Here Hab[A] is an SO(4,1) intertwiner and tangential,(5.84)Habxa=0,H≔ηabHab=12LNC4ðaAa. Then the linearized effective metric (5.81) becomes in SO(4,1)-covariant notation(5.85)Gab=Pab+h˜ab,withh˜ab≔(hab−12Pabh), thus defining the physical graviton h˜ab, where(5.86)hab=4LNC4[Hab]0,h=ηabhab is dimensionless. We study the graviton modes (5.85) for the spin s=0,1,2 fluctuations of (5.29) in more detail below.5.6.1Spin 0 gravitonsTo begin with, consider the perturbation (5.83) of the metric for the two spin 0 modes of (5.11). One finds(5.87)Hab[A(1)]=θacθbd(ðcðdϕ(0)+ðdðcϕ(0)),Hab[A(2)]=r2(xaθbdðdϕ(0)−R2θadðdðbϕ(0))+(a↔b). Upon averaging, one obtains(5.88)hab[B(1)]=α1(2Pabð⋅ðϕ(0)−(∇a∇bϕ(0)+∇b∇aϕ(0))),hab[B(2)]=0, and the expressions satisfy(5.89)h[B(1)]=6□ϕ,∇ahab[B(1)]=0, using ∇ahab=ðahab−1R2xbh. Then the physical graviton of (5.85) satisfies the de Donder gauge,(5.90)∇ah˜ab[B(1)]−12∇bh˜=0withh˜ab[B(1)]=hab[B(1)]−12Pabh˜. 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.5.6.2Spin 1 gravitonsNext, we compute the spin one contributions to the gravitons on H4. Taking into account the Cs gradation, the averaged metric perturbation (5.83) is non-vanishing only for the modes Aa(3) and Aa(2′).Spin 1 graviton Aa(2). Here, we observe(5.91)Hab[A(2)]=−r2R2{Pab,ϕ(1)}−r2R2(∇aAb(2)+∇bAa(2))+r2(xaAb(2)+xbAa(2))=−r2R2(∇aAb(2)+∇bAa(2)) such that the averaging yields(5.92)hab[A(2)]=α1(∇a(□−2r2)ϕb+∇b(□−2r2)ϕa). 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.Spin 1 graviton Aa(3). Similarly, we have(5.93)Hab[A(3)]=θadθbf(ðfϕd+ðdϕf) such that averaging yields(5.94)hab[A(3)]=−α1(∇aϕb+∇bϕa).Physical spin 1 gravitons. For the spin 1 eigenmodes B(i) of (5.47), we therefore obtain the following physical gravitons:(5.95)h˜ab[B(1)]=h˜ab[B(4)]=0,h˜ab[B(2)]=α1(1−α1)(∇a(□−2r2)ϕb+∇b(□−2r2)ϕa),h˜ab[B(3)]=−α1(∇aϕb+∇bϕa). 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 Tμν.5.6.3Spin 2 gravitonsFinally, we consider the spin 2 fluctuations of the background and evaluate their associated graviton modes.Spin 2 graviton Aa(1). Since Aa(1)=ðaϕ(2) with ϕ(2)={xa,{xb,ϕab}}∈C2, we have(5.96)Hab=θda{ðdϕ(2),xb}+(a↔b)={xb,{θadðdϕ(2)}−{θda,xb}ðdϕ(2)+(a↔b)={xb,{xa,ϕ(2)}}−{θda,xb}ðdϕ(2)+(a↔b). The second term drops out in the projection to C0, and using (4.29) twice one finds(5.97)hab[A(1)]=4LNC4[{xb,{xa,ϕ(2)}}]0=2R2r2αˆ1αˆ2ϕab.Spin 2 graviton Aa(2). For Aa(2)={xc,ϕbc}∈C1⊕C3, it follows that Hab∈Codd and therefore(5.98)hab∝[Hab]0=0. In fact this is a pure gauge mode in the model.Spin 2 graviton Aa(3). Next, consider Aa(3)={xc,ϕac}∈C1. Then Hab∈Codd, and again(5.99)hab∝[Hab]0=0.Spin 2 graviton Aa(4). Finally, consider the mode Aa(4)=θae{xc,ϕec}∈C2. Then(5.100)Hab=θda{θde{xc,ϕec},xb}+(a↔b)=θdaθde{{xc,ϕec},xb}+θda{θde,xb}{xc,ϕec}+(a↔b)=r2R2{{xc,ϕac},xb}−r2θba{xc,ϕec}xe+(a↔b)=−r2R2{xb,{xc,ϕac}}+(a↔b) using (4.15). Recall that (4.29) implies [{xb,{xc,ϕac}]0=−αˆ1ϕab, and therefore(5.101)hab[A(4)]=2α1(□−2r2)ϕab.Physical gravitons. Computing the gravitons for the eigenmodes B(i), we find(5.102)h˜ab[B(i)]=0fori=1,2,3h˜ab[B(4)]=2α1(□−2r2)ϕab,ðah˜ab[B(4)]=0=∇ah˜ab[B(4)] using (D.26). The trivial result for B(i), i=2,3, is obvious, as the individual contributions for A(i), i=2,3, vanish. However, the vanishing contribution of B(1) is the result of a non-trivial cancellation of the contributions from A(1) and A(4).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 hab decompose in general as(5.103)hab=hab(2)+∇aξb+∇bξa+14ηabh where hab(2) 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 ξa contains another spin 0 (divergence) mode. While the ξa 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. [35]. This does not happen here, which is certainly welcome for the quantization of the model.5.7Classical action for metric fluctuationsHaving 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:(5.104)S2=1g2∫ρB(4)D2B(4)=1g2α1r2R2∫ρϕab(2)[B(4)](□−2r2)2ϕab(2)[B(4)]=14α1gYM2LNC4∫h˜ab[B(4)]h˜ab[B(4)] where gYM 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.1717Hence a large positive mass would not imply large curvature but rather a short range of these modes. See e.g. [36] for a related discussion.Taking into account a coupling to matter of the form δhS=12∫h˜abTab, the equations of motion for h˜ab become(5.105)h˜ab[A(4)]=−43g2ρLNC4Tab=−13gYM2LNC4Tab. Clearly h˜ab is not propagating, but acts like an auxiliary field which tracks Tab. As a consequence, the pure matrix model action (5.1) does not lead to gravity on H4, similar to the case of SN4 [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 ϕab, which should be suitable for quantization. Gravity may then arise upon quantization, as discussed next.5.8Induced gravityAt 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.1818In fact, it is known that the Type IIB bulk gravity in the IKKT model arises only at one loop [37]. However, this is a different issue, since the present degrees of freedom are only 4-dimensional.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 [17,18,38]. The induced terms include the cosmological constant and Einstein-Hilbert terms. The maximal supersymmetry of the underlying model1919This really requires the maximal supersymmetry of the IKKT model, otherwise UV/IR mixing effects will render the model strongly non-local and probably pathological, cf. [39,40]. 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 ∫gΛ4 is avoided here, cf. the one-loop computation in [3]. Canceling also the induced Einstein-Hilbert term is more subtle,2020On Moyal–Weyl backgrounds, N=1 SUSY is sufficient to cancel the induced “would-be” cosmological constant term, while the induced Einstein–Hilbert term is only canceled in the N=4 case [41,42]. This is reflected by the absence of UV/IR mixing. Here the background and the explicit mass term induce a spontaneous and soft breaking of N=4 SUSY. Nevertheless, the suggested scenario seems reasonable. and it is plausible that the supersymmetry breaking H4 background does lead to an induced Einstein-Hilbert term with scale Λ˜=O(1r).Motivated by these considerations, one may add a term ∫σΛ˜2h˜abð⋅ðh˜ab to the action (5.104), with σ=±1, such that the total action coupled to matter reads(5.106)S=∫σΛ˜2h˜abð⋅ðh˜ab+43gYM2LNC4∫h˜abh˜ab+12∫h˜abTab. The equation of motion for h˜ab are then(5.107)(ð⋅ð+43σgYM2L4NCΛ˜2)h˜ab=−14σΛ˜2Tab where Λ˜ 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(5.108)8πGN=18Λ˜2 and mass scale(5.109)m2=O(1gYM2LNC4Λ˜2). The mass scale can become very small m2=O(1R2) if Λ˜=O(1r) and n is large, or upon projection to the Minkowski space-time M3,1, 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 6.5.9Local gauge transformationsAmong the higher-spin gauge transformations δΛ(xa+Aa)≔{xa+Aa,Λ} generated by Λ∈C, consider the spin 1 gauge transformations generated by(5.110)Λ(1)={xa,va}=θabðbva∈C1 with va(x) a divergence-free vector field. These correspond to (volume-preserving) diffeomorphisms on H4. The action on scalar functions ϕ(x) reads(5.111)δΛϕ={ϕ(x),Λ(1)}={ϕ(x),θabðbva} so that the action on vector fluctuations is(5.112)δΛAa=δΛxa+{Aa,Λ(1)} with2121Note that {Aa,Λ(1)} is not necessarily tangential. However that term vanishes in the semi-classical limit, and is significant only for nonabelian gauge fields which we do not consider. The proper treatment is of course to impose the non-linear constraint YaYa=−R2, which would restore gauge invariance.(5.113)δΛxa={xa,Λ(1)}={xa,Λ}0+{xa,Λ(1)}2=α1(□−2r2)va+Aa(2′)[Λ(1)] using (4.9). The first term describes a diffeomorphism corresponding to the vector field v˜a=α1(□−2r2)va. The second term accounts for the spin 1 pure gauge mode Aa(2′) 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(Hn), see (3.55).6Lorentzian quantum space-times from fuzzy Hn4Having disentangled the fluctuations on Hn4, we would like to apply these tools to the more interesting cosmological space-time solutions M3,1. 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(Hn) such that we can rely on the same spin operator S2, and our classification can be carried over. Moreover, the tangential fluctuations on Hn4 are in one-to-one correspondence to the full set of fluctuation modes on M3,1, 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.6.1Cosmological space-time solutionsBy projecting fuzzy Hn4 onto the 0123 plane via Π of (2.30) i.e. by keeping the Yμ=Mμaαa for μ=0,1,2,3 and dropping Y4, we obtain (3+1)-dimensional fuzzy space-time solutions. Since the embedding metric ημν is compatible with SO(3,1), we have(6.1)[Yρ,[Yρ,Yμ]]=i(α⋅α)[Yρ,Mρμ]=−i(α⋅α)[Mρμ,Yρ]=(α⋅α){Yμ,μ≠ρ0,μ=ρ(no sum) such that(6.2)□YYμ=[Yρ,[Yρ,Yμ]]=3(α⋅α)Yμ. 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:(6.3)□XXμ=−3r2Xμ,□TTμ=3R2Tμ,□ZZμ=0. Choosing a positive mass term to ensure stability, we focus on the solution(6.4)Yμ=Xμ,r2=13m2. This is the homogeneous and isotropic quantized FLRW cosmological space-time Mn3,1 with k=−1 introduced2222We change notation from [1], where Y1 was dropped instead of Y4. in [1]. Here m2 sets the scale r2, while n remains undetermined. These backgrounds are SO(3,1)-covariant, which is the symmetry respected by □X.6.2Semi-classical geometryWe first recall the semi-classical limit of this space [1], with xμ for μ=0,1,2,3 as coordinates on M. By SO(3,1)-invariance, we can always consider the local reference point ξ on H4 resp. M(6.5)ξ=(x0,0,0,0,x4)→Π(x0,0,0,0),x0=Rcosh(η),x4=Rsinh(η). Globally, we have the following constraints(6.6)xμxμ=−R2−x42=−R2cosh2(η),tμtμ=r−2cosh2(η),tμxμ=0,μ,ν=0,…,3 where η will be a global “cosmic” time coordinate. From the radial constraint xaxa=−R2 on H4 one deduces {xaxa,xμ}=0, which further implies(6.7)0=xamaμ=xνmνμ+x4m4μ. This establishes a relation between the momenta and the tμ,(6.8)tμ=1Rmμ4=1Rr2x4xνθνμ=ξ1Rr21tanh(η)θ0μ. Furthermore, the self-duality constraint (3.58c) reduces to2323Note that this form only applies in the special so(3,1) adapted frame, and it is not generally covariant; of course on Minkowski manifolds, there is no notion of self-duality. However there can be a SO(3,1)-invariant relation as above which holds in the preferred cosmological frames, and this is what happens here. This is one reason why it is important to not have full Poincare covariance in the Minkowski case.(6.9)ti=1Rmi4=1nRr3ϵabci4θabxc=ξ1nr3cosh(η)ϵijkθjk,t0=ξ0, where the last equation is simply a consequence of xμtμ=0. Therefore tμ describes a space-like S2 with radius r−2cosh2(η). Conversely, the above relations allow to express θμν in terms of the momenta tμ as follows(6.10)θij=nr32cosh(η)εijktk,θ0i=Rr2tanh(η)ti. By means of R∼12nr, one can summarize (6.10) neatly:(6.11)θμν=r2Rηαμν(x)tα, where ηαμν(x) is a SO(3,1)-invariant tensor field on M3,1, which is analogs of the t'Hooft symbols. Note that θ0i≫θij for late times η≫1; this reflects the embedding of H4⊂R4,1 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 S2. Finally the constraint (3.58d) reads(6.12)γαβ≔ημνθμαθνβ=LNC44(ηαβ+1R2xαxβ−R2tαtβ) which at the chosen reference point yields(6.13)γij=LNC44(δij−R2titj),γ00=LNC44sinh2(η),γ0j=0.Averaging and effective metric on M3,1. An effective metric for scalar fields ϕ(x) on M3,1 can be defined by the quadratic action (5.79). Looking at (6.12), we note that γαβ contains the term tαtβ, which is not constant on the fiber S2. By averaging over the fiber, one obtains the following result [1](6.14)[γij]0=LNC44δij−[titj]0=LNC412(3−cosh2(η))δij,[γ00]0=LNC44sinh2(η),[γ0i]0=0. Note the signature change at cosh2(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 Gμν (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 Gμν 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.6.3Wave-functions, higher-spin modes and constraintsIn this section we briefly comment on the fluctuation modes on M3,1. The space of functions End(Hn) on M3,1 is the same as on Hn4, 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 H4, such that a representation as in (3.42) is expected to hold. Consequently, the modes can be interpreted as functions (or higher-spin modes) on2424We will ignore the dependence on two sheets M± for simplicity. M3,1 via Π of (2.30). The ϕab(x) etc. then define some higher-rank field on M3,1. In the following, we will only address a few basic points.6.4Tangential fluctuation modes, relation with H4 and SO(4,1)Now consider fluctuations yμ=xμ+Aμ around M3,1. The first observation is that these four fluctuation modes Aμ,μ=0,…,3 are in one-to-one correspondence with the tangential fluctuations on H4. To see this, recall that tangential fluctuations on H4 satisfy by definition2525A gauge-invariant constraint would be (Xa+Aa)(Xa+Aa)=−R2. For the present purpose, its linearized form is what we want. the constraint(6.15)Aaxa=0,A4=−xμx4Aμ, with Aa∈End(H). To associate a general fluctuation mode on M4,1 one simply drops A4, and conversely A4 can be recovered from Aμ via (6.15). Hence there is a correspondence of tangential fluctuations(6.16) Since the maps are invertible, an SO(4,1)-action is defined on the fluctuations Aμ on M3,1, which, however, is not an isometry and not unitary. Nevertheless it acts as a structural group, and organization developed for H4 in the previous sections remains applicable. As a consequence, configurations in the M3,1 model can be mapped one-to-one to configurations in the H4 model. Similarly, higher-rank tangential tensors on H4 such as the gravitons(6.17)habxa=0 can be mapped one-to-one to tensors hμν on M3,1, and the missing components hab are uniquely determined from the hμν. In the same vein, all internal fluctuations on S2 will be organized in a SO(4,1)-covariant way as on H4. This relation is somewhat analogous to a Wick rotation.Action and dynamics. The matrix model provides again an action for the fluctuation modes Aμ, which has the same structure as in section 5.4,(6.18)SM[A]=∫Aμ(□M−2I+12μ2)Aμ upon gauge fixing. The matrix Laplacian on M3,1 is related to the one on H4 through(6.19)□M=□H−[X4,[X4,⋅]]=[Xμ,[Xμ,⋅]]∼−LNC44γμν∂μ∂ν+…. We can utilize the same mode expansion in terms of A(i)μ as in section 5.1,(6.20)Aμ(1)=ðμϕ(s)∈Cs,ϕ(s)={xa,ϕa(s)}={xμ,ϕμ(s)}+{x4,ϕ4(s)}∈CsAμ(2)=θμbðbϕ(s)={xμ,ϕ(s)}∈Cs+1⊕Cs−1Aμ(3)=ϕμ(s)∈Cs−1,Aμ(4)=θμbϕb(s)∈Cs, 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 M3,1 without changing the setup. The relation (4.9) still applies; for example(6.21){xμ,ϕ(1)}0=−23ϕμ+R23ðcðcϕμ. Note that ðcðc is the Euclidean Laplace operator on H4, 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 ϕa1…a2 amount to field redefinitions. Therefore one should expect that these higher-derivative 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. {xa,ϕa(s)} in (6.20) is replaced by {xμ,ϕμ(s)}. However then some identities are lost, and it remains to be seen which formalism is more advantageous.7Conclusion and outlookIn this article we provide a careful and detailed analysis of the fluctuation modes on fuzzy Hn4 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 SN4 [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 Hn4 is identified as the minireps or doubleton Hn, for which we recall the oscillator realization in section 3. The first major step is a classification of the fuzzy algebra of functions End(Hn), which relies on two pillars: (i) the construction of a spin Casimir invariant S2<