2D fuzzy Anti-de Sitter space from matrix models

We study the fuzzy hyperboloids AdS^2 and dS^2 as brane solutions in matrix models. The unitary representations of SO(2,1) required for quantum field theory are identified, and explicit formulae for their realization in terms of fuzzy wavefunctions are given. In a second part, we study the (A)dS^2 brane geometry and its dynamics, as governed by a suitable matrix model. In particular, we show that trace of the energy-momentum tensor of matter induces transversal perturbations of the brane and of the Ricci scalar. This leads to a linearized form of Henneaux-Teitelboim-type gravity, illustrating the mechanism of emergent gravity in matrix models.


Introduction
There has been a great amount of work on noncommutative field theory on the the fuzzy sphere and similar compact quantum spaces. Part of their appeal stems from the fact that the space of functions on these spaces has a simple group-theoretical structure and is finite-dimensional, reflecting their finite symplectic volume. This leads to mathematically well-controlled toy models for noncommutative field theory and geometry, see e.g. [1][2][3][4][5][6][7][8][9] and references therein. However, most of the work so far has been for spaces with Euclidean signature, and it would be desirable to know more about fuzzy spaces with Minkowski signature.
In this paper, we study in detail 2-dimensional fuzzy de Sitter space dS 2 and Anti-de Sitter space AdS 2 , which are quantized homogeneous spaces with Minkowski signature and non-vanishing curvature. Fuzzy AdS 2 has been studied previously in [10,11]. In the first part of this paper, we elaborate the space of functions on these fuzzy hyperboloids, and provide explicit formulae for the square-integrable wavefunctions corresponding to unitary irreducible representations of SO (2,1). For the discrete series representations we recover previous results obtained in [10], and for the principal continuous representations our results are new. This provides the basic constituents for quantum field theory 3 on fuzzy AdS 2 and dS 2 . In particular, this also allows to establish the required quantization map for the fuzzy geometry.
In a second part, we consider a matrix model which describes dynamical fuzzy AdS 2 and dS 2 spaces as brane solutions. As discussed in [13], this leads to a dynamical effective geometry on the branes, determined by a combination of the embedding geometry of the brane and its Poisson structure. The present 2-dimensional example provides an interesting toy model for emergent gravity, with a non-trivial curvature background. We study the perturbations around the AdS 2 solutions, and their dynamics in the presence of matter. This is interesting because the extrinsic curvature of the brane leads to a coupling of the linearized matrix perturbations to the energy-momentum tensor 4 , as pointed out in [14,15]. More precisely, the transversal perturbations of the brane couple to the trace of the energy-momentum tensor of matter, due to the extrinsic curvature. It turns out that the perturbations of the effective metric are governed by a linearized Henneaux-Teitelboimtype gravity [16], relating the trace of the energy-momentum tensor to the Ricci scalar. This is remarkable, because it results directly from the underlying matrix model action, without adding any gravity action. It provides a simple example for the mechanism of emergent gravity in Yang-Mills matrix models. However, this result is restricted to the linearized regime.
In 4 and higher dimensions, the dynamics of the effective geometry is complicated due to a mixing between tangential and transversal brane perturbations [14], which prohibits a full understanding at present. A similar mixing is observed here, but we are able to disentangle the coupled wave equations, and thereby essentially solve the perturbative dynamics. Therefore the present 2-dimensional case should serve as a useful step towards understanding the more complicated higher-dimensional case.

.1 Geometry and isometry group
There are three types of two-dimensional non-compact spaces with constant curvature, given by the Anti-de Sitter space AdS 2 , de Sitter space dS 2 and the hyperbolic or Lobachevsky plane H 2 . In this paper we discuss AdS 2 and dS 2 , which can be naturally realized as the one-sheeted hyperboloid embedded in R 3 through In terms of conformal coordinates −π/2 < σ < π/2 and π < τ ≤ π, the embedding of classical Anti-de Sitter space AdS 2 is given by The induced metric is pseudo-Riemannian with closed time-like circles 5 around x 3 = const . De Sitter space dS 2 is obtained from AdS 2 by switching the roles of the time and space, thus changing the overall sign in the metric. The circles x 3 = const are then space-like, and there are no closed time-like curves. Both AdS 2 and dS 2 admit the group SO(2, 1) or its cover SU(1, 1) as isometries, generated by vector fields K a , a = 1, 2, 3 which close su(1, 1) Lie algebra with respect to commutators or explicitly The Casimir operator of su(1, 1) Lie algebra is defined as As usual, it is convenient to introduce the ladder operators which satisfy the commutation relations Then unitary irreducible representations of SO(2, 1) are spanned by a basis |j, m of weight states, where j is related to the eigenvalue of the Casimir C, and m is the eigenvalue of K 3 and the action of K ± on |j, m produces a state with weight m ± 1: A chain of states obtained by the successive action of K − operator terminates if there exist state such that K − |j, m 0 = 0.
Denoting this lowest weight by j = m 0 , it follows that Therefore the chain of states which span this irreducible lowest weight representation is determined by the state |j, j of lowest weight, via By analogy, the highest weight representation are obtained by interchanging roles of K + and K − operators. If no lowest or highest weight state exists, then the normalisability condition implies C < 0, and the states belong to the unitary irreducible continuous representations. In general, the resulting structure of irreducible representations is as follows: where The finite-dimensional irreducible representations of SU(1, 1) are obtained for j ∈ −N/2. They are not unitary, and correspond to the spin |j| representations V |j| of SU(2) with C = −|j|(|j| + 1). All unitary irreducible representations are infinite-dimensional, and fall into one of the following classes 6 [17]: • The discrete series of the highest and the lowest weight representations characterized by C = j(j − 1) ≥ 0.

Functions and Poisson bracket
In order to carry out the quantization of (A)dS 2 , it is useful to organize the space of functions on (A)dS 2 in terms of irreducible representations of SU (1,1). This provides at the same time the basis of eigenfunctions of the invariant d'Alembertian g , 6 We only consider representations with integer weights for simplicity.
which is related to the Casimir operator of su(1, 1). Here g = det(g µν ), and g µν is inverse of the metric. We can thus decompose any function on the hyperboloid into eigenfunctions of g , and label the solutions by j and m as above. The solutions corresponding to the finitedimensional representations are realized by polynomial functions Pol(x a ); they are of course not normalizable on (A)dS 2 . The square-integrable functions corresponding to unitary irreducible representations are given explicitly in terms of hyper-geometric functions is the Casimir. For AdS 2 , the scalar fields corresponding to positive or negative energy unitary representations belong to the discrete representation D ± j , with α > 0. Then the equations for the lowest weight state have a unique solution and the spectrum is non-degenerate. On the other hand the states given by (22) with α < 0 belong to the continuous representations, with two-fold degenerate spectrum. These are the physical scalar fields on de Sitter space dS 2 . Putting these together, we have the following decomposition of functions on the hyperboloid (A)dS 2 along with the space of polynomial functions Pol(x a ). In the following we discuss fuzzy versions of these non-compact spaces, and their associated spaces of functions. As a starting point, we note that the natural SO(2, 1)-invariant volume element endows the hyperboloid with a non-degenerate symplectic form with dω = 0, introducing a scale perameter κ. Its inverse defines the Poisson bracket of two functions We can now look for a quantization of this Poisson manifold M, cf. [20]. This means that the algebra of functions C(M) should be mapped to a non-commutative (operator) algebra A, such that the commutator is approximated by the Poisson bracket. In the present case, the group-theoretical structure of (A)dS 2 provides a natural and explicit quantization, in analogy to the case of the fuzzy sphere [1]. As a first step, we note that the Poisson brackets of the embedding functions x a satisfy the Lie algebra of SO(2, 1) where f ab c are structure constants of SO(2, 1). This implies as usual that the corresponding Hamiltonian vector fields satisfy the same Lie algebra, and indeed it is easy to verify that the SO(2, 1) vector fields (5) are given by 3 Fuzzy hyperboloid In analogy to the fuzzy sphere [1], we define fuzzy two-dimensional hyperboloid in terms of three hermitian matrices (or operators) X a , which are interpreted as quantization of the embedding functions x a . In view of (28), we impose the following relations where f ab c are structure constants of the Lie algebra su(1, 1). Therefore the X a are rescaled su(1, 1) generators, and we assume that they act on a certain irreducible unitary representation H j of the Lie algebra. We can then write the Casimir operator as Since H j is assumed to be irreducible, the X a generate the full algebra A of operators on where H * j is dual representation of H j . This algebra is an infinite-dimensional vector space, which naturally carries an action of su(1, 1) by conjugation with the generators X a : We now specify the representation H j . Since the matrices X a should be interpreted as quantized embedding functions x a of the hyperboloid and comparing the spectrum of X 3 with the range of x 3 ∈ −∞, ∞ , we choose H j to be a principal continuous representation 7 , in accord with [10]. We can furthermore define an invariant scalar product A contains in particular the polynomials generated by the X a , where this trace diverges. However, A also contains normalizable matrices corresponding to physical scalar fields, which are of main interest here. Finding such normalizable matrices is equivalent to decomposing A = H j ⊗ H * j into irreducible unitary representations of su(1, 1). This problem has been extensively studied in the literature [18,19]. In general, the states |JM which belong to a particular unitary irreducible representation in H j 1 ⊗ H j 2 are given by Here the C's are the Wigner coefficients, which vanish unless M = m 1 + m 2 . In the special where the D's vanish unless M = m 1 − m 2 ∈ Z. Since we chose the principal continuous representation H j ∼ = P s , one obtains the following decomposition of the space of functions A into unitary modes [18]: along with the space of polynomial functions Pol(X a ). In the next section we will recover this result and obtain the corresponding fuzzy wavefunctions explicitly, which solve the eigenvalue equations

Fuzzy wavefunctions
We can determine the fuzzy wavefunctions Φ J M explicitly, using their definition as irreducible representations of SO(2, 1). As an element of the operator algebra A, the matrix Φ J M acts on |jn ∈ H j as Defining the matrix D J M (K 3 ) by its action on |jn and using defining property Γ(x + 1) = xΓ(x) of Gamma function, we can express Φ J M for integer M using (15) as To derive the final expressions (41) and (42) one applies the identity which can be verified using which follows from (10). We note that the expressions (41) and (42) (37) with J being integer, the basis of states is completely determined by the minimal weight state annihilated by K − . Acting with K − on (41) we obtain Specializing this to the case M = J, we see that the expression in square bracket must vanish Since K 3 takes only integer values here, we can conclude that (up to normalization). Finally, for the lowest weight state in D + J using (43) we obtain in agreement with findings of [10,11]. The highest weight state in D − J is given by hermitian conjugate of (48).
For the principal continuous representation P S with J = 1/2 + iS in (37), the matrices D J M are solutions of second order difference equation obtained from (45) applying K + on it: Here we find it convenient to write D J M as where C J M (K 3 ) is a matrix with elements given by the Wigner coefficients for the principal continuous representations of su(1, 1) in (37). This can be seen after noting that this second order difference equation is a special case of equation for the general Wigner coefficients as found in [19]. Finally, we can express C J M in terms of two independent solutions 8 .
defined by To summarize, we have obtained explicit matrices Φ J M of the form realizing the decomposition (37) of A = H j ⊗ H * j into unitary representations of SO(2, 1). They form an orthonormal basis for the inner product defined by the trace (34). This is in one-to-one correspondence with the decomposition (25) of classical functions with x + = x 1 +ix 2 on the hyperboloid. Due to the relation with the Casimir, the spectrum of matrix d'Alembertian 1 κ 2 in (38) and the classical d'Alembertian (20) coincide. Including also the space of polynomial functions Pol(X a ), this is the basis for interpreting the matrix algebra A as quantized algebra of functions over hyperboloid.

Semi-classical limit
Now consider the classical hyperboloid M as a Poisson manifold equipped with the Poisson structure (27). The quantization of such a Poisson manifold is defined in terms of a quantization map Q, which is an isomorphism of vector spaces from the space of functions C(M) on M to some (operator) algebra A which is compatible with the Poisson structure {f, g} = θ µν ∂ µ f ∂ ν g, satisfying Clearly Q ≡ Q θ depends on the Poisson structure θ, and the limit θ → 0 is understood in some appropriate way; for a more mathematical discussion we refer e.g. to [20]. As Q is an isomorphism of vector spaces 9 , one can then define the semi-classical limit of some fuzzy wavefunction F ∈ A as the inverse f = Q −1 (F ) of the quantization map. This is consistent as θ → 0, provided commutators are replaced by the appropriate Poisson brackets, and higher order terms in θ are neglected.
In general, there is no unique way of defining Q. However in the case at hand, there is a natural definition of Q, based on the decomposition of C(M) and A into irreducible representations of SO(2, 1). Given the corresponding orthonormal bases Φ J M and φ J M of A resp. C(M) as obtained above, we define so that Q is an isometry for the unitary representations. This can be extended to the polynomials Pol(x a ), corresponding to finite-dimensional non-unitary irreducible representations of SO(2, 1). However these are not normalizable, and the normalization of Q(Pol(x a )) must be fixed in another way. Since we want to interpret the matrices X a , a = 1, 2, 3 as a quantized embedding coordinates x a , a = 1, 2, 3, we define Comparing the embedding equation x a x a = −R 2 with the Casimir constraint X a X a = κ 2 j(j − 1) (31), we are led to impose using j = 1/2 + is for the principal continuous representation H j . Therefore the semiclassical limit κ → 0 implies 10 s → ∞. This is the analog of N = dim H → ∞ in the case of fuzzy sphere.
To establish 11 the required properties of Q, we recall that all fuzzy wavefunctions can be written in the following "normal form" (54) and similarly for M < 0. We claim that as functions in one variable. To see this, observe that in the limit κ → 0 following relations hold as a consequence of the Lie algebra relations. In the classical case, the corresponding relations are Therefore the action of the matrix Laplacian (38) on Φ J M in the limit κ → 0 has precisely the same form as the action of the classical Laplacian This implies (63) up to normalization, and allows to define Q in such a way that In particular, this provides the appropriate definition of Q for the principal continuous representation P S (which is doubly degenerate), as well as for the finite-dimensional polynomials which are not normalizable. Now it is easy to see that Q respects the algebra structure and the Poisson bracket in the limit κ → 0. Consider the product of two matrix modes as above, expanded up to leading order in κ for M, M ′ ≥ 0. Then (57) follows immediately using (63). Subtracting the same computation with the factors reversed, (58) follows. A similar computation applies to modes with mixed or negative M. Therefore Q is indeed a quantization map for our Poisson structure. Using the decomposition of A into the above modes, analogous statements can be made for the de-quantization map Q −1 , which provides the semi-classical limit of the fuzzy wavefunctions. Finally, consider the trace of some normalizable wavefunctions with weight M = 0, using the explicit form of H j = P s (18), where ω is the symplectic form (26). This computation is easily generalized to show that as long as the integrals are bounded. This is guaranteed for the spaces of unitary wavefunctions discussed above. To summarize, in the semiclassical limit ∼ defined as de-quantization map expanded up to leading order in κ, we can use the following relations which we use in the following sections.

Dynamical matrix models
Consider now three hermitian matrices X a = (X a ) † ∈ Mat(∞, C) for a = 1, 2, 3, which transform in the basic 3-dimensional representation of SO(2, 1). Then the most general matrix model up to order 4 which is invariant under the SO(2, 1) symmetry as well as translations X a → X a + c a 1l has the form for suitable constants, where embedding indices are raised and lowered with η ab . The matrices X a are understood to have dimension length, and accordingly [g Y M ] = L 2 . This model is invariant under SO(2, 1) rotations, translations as well as gauge transformations X a → UX a U −1 for unitary operators U. The equations of motion are obtained as Now consider the ansatz 12 X a = κK a , a = 1, 2, 3 in terms of rescaled generators of a unitary irreducible representation of SO(2, 1). Then where C is the quadratic Casimir of SO(2, 1), and κ, R are positive numbers. As discussed before, we take H = H j to be the principal continuous series representation, so that X a ∈ End(H) and Thus the equations of motion (75) are solved by this ansatz provided This is a quadratic equation in κ which we assume to have a positive solution. Let us discuss the geometry of the fuzzy brane solutions in the matrix model in the semi-classical limit, following [13]. Recall that the matrices X a are interpreted as quantized Cartesian embedding functions The induced metric on M is given by For the hyperboloid solutions under consideration x a x a = −R 2 holds, so that the induced metric is that of AdS 2 . However, we are interested in the effective metric which governs physical fields in the matrix model. To identify the effective metric in the semi-classical limit, we note that the kinetic term (with two derivatives) for e.g. scalar fields Φ in the matrix model 13 arises from an action of the form using the semi-classical correspondence rules (73). Here the scalar fields are made dimen- Therefore the effective metric is given by G µν . Note the explicit minus in the definition of G µν , which is in contrast to the higher-dimensional case discussed in [13]. The correct sign is dictated by the action (74) resp. (82), which must have the form S = dt(T − V ). For the action (74) it means that the effective metric is indeed that of AdS 2 , while fuzzy dS 2 can be obtained by changing the overall sign of the action. This choice of signs is possible only in the case of signature (−+) in 2 dimensions. Note also that for 2-dimensional branes, the conformal factor of the effective metric is not fixed by the above scalar field action, due to the Weyl symmetry G µν → e α G µν . Here we choose (83) for simplicity; our main goal is to illustrate how such an effective metric responds to matter perturbations in the present matrix model. The relation G ∼ g is particular for 2 dimensions, and can be seen in coordinates where g µν = diag(−1, 1) at some given point p ∈ M 2 . Consider the point p N = (R, 0, 0) in the homogeneous AdS 2 space. Its tangent space is parallel to the (x 2 x 3 ) plane, so that we can use x µ = (x 2 , x 3 ) as local coordinates. In these "normal embedding" coordinates we have g µν = diag(−1, 1) at p N , and θ 23 = {x 2 , x 3 } = κf 23 1 R = κR. On the other hand θ µν = {x µ , x ν } = g Y M e −σ/2 ǫ µν using (83), and we obtain We note that the matrix Laplace operator (77) for the unperturbed hyperboloid background is related to the geometric Laplace operator in the semi-classical limit 14 Finally, it is easy to add fermions the matrix model, via the action Here is a 2-component spinors of SO(2, 1), and Γ a satisfy the Clifford algebra of SO(2, 1),

Fluctuating AdS 2 and gravity
We introduce some useful geometrical structures which apply to general M 2 ⊂ R 3 . The "translational currents" span the tangent space of M 2 ⊂ R 3 , while characterizes the extrinsic curvature and is normal to the brane with respect to the embedding metric, In particular, is a normal vector 15 to M 2 ⊂ R 3 . For the present AdS 2 solution, one can easily compute the curvature of the connection 14 Although such a relation holds very generally in the higher-dimensional case [13], it is restricted to e σ = const in 2 dimensions; for a general formula in 2 dimensions see [21]. Here we need the Laplacian only for the unperturbed backgrounds, where (85) is sufficient. 15 In general, this holds for g rather than G , but in the 2-dimensional case both statements are true. This is consistent with g x a = 2 R 2 x a on AdS 2 . The Riemann curvature tensor can be obtained e.g. from the Gauss-Codazzi theorem, and is given by using (84), and recalling x a x a = −R 2 . Using the above relations along with (77), the embedding functions x a satisfy Now consider small fluctuations around the solutionsX a of the above matrix model, parametrized as X a =X a + A a (X) .
These fluctuations can be interpreted in different ways. First, one can decompose the A a into tangential and one radial components, analogous to the well-known case of the fuzzy sphere [6]. Then the radial component can be interpreted as scalar field on M 2 , and the tangential components in terms of (noncommutative) gauge fields. This interpretation is certainly appropriate for the non-abelian components, which arise on a stack of n coinciding such branes. However since the trace-U(1) components change the effective metric G µν on M 2 , it is more natural to interpret them in terms of geometrical or gravitational degrees of freedom; note that there is no charged object under this U(1). In this section we elaborate some aspects of the resulting 2-dimensional effective or "emergent" gravity 16 . In the semi-classical limit, the matrix model action expanded to second order in A a around the basic AdS 2 solution is given by dropping the linear as well as the f abc term which vanish due to the equations of motion (79), and using Here can be viewed as gauge fixing function, since it transforms as under gauge transformations. We can thus choose the gauge such that f = 0. We want to understand how the geometry is influenced by matter. We assume that all fields on M 2 couple to the effective metric 17 G µν , so that the metric perturbations couple to matter via the energy-momentum tensor. The linearized metric fluctuation is given by where we decompose the perturbations into tangential and transversal ones Using 2K a µν = e σ g µν K a (93), the perturbation of the effective metric in Darboux coordinates can be written as Therefore noting that ∇θ µν = 0, where T = T µν G µν , and we definẽ for convenience. Thus the normal component A ⊥ couples to the trace of the energymomentum tensor, while the tangential components couple to its derivative. This illustrates the observation [14] that a non-derivative coupling of the embedding perturbations to the energy-momentum tensor arises on branes with extrinsic curvature. Using the on-shell condition (79) for the background and we obtain the semi-classical equations of motion Note that the normal component couples to T via the extrinsic curvature. This is the crucial ingredient for gravity, as we will see below.

Curvature perturbations and gravity
Now we can obtain the curvature perturbations induced by matter. Since in 2 dimensions Ric µν [G] = 1 2 G µν Ric [G] where Ric[G] is the Ricci scalar, we will restrict ourselves to study the linearized perturbations of Ric [G]. This can be computed using which implies The perturbation of the effective metric in Darboux coordinates can be written as follows (cf. (103)) using δθ µν = 0. After some computations given in the appendix, the corresponding perturbation of the Ricci tensor is obtained as This can be seen as linearization of the following gravity model which is reasonable and non-trivial in 2 dimensions [16] (dropping the O(∂∂T ) terms), unlike general relativity which does not allow any coupling to matter. Note that the derivative term is of order using (106), and can be neglected provided e σ ≪ 1, which is compatible with G N ≪ 1.
Although we focused on the AdS 2 background, the result should equally apply to the dS 2 background, which is obtained by changing the sign of the matrix model action.
We emphasize again that no specific gravity action was assumed or induced, we have simply elaborated the matrix model dynamics from a geometrical point of view. The crucial coupling to T µν arises due to the extrinsic curvature of the brane encoded in ∇ µ J a ν = K a µν , as pointed out in [14]; this is already seen in (107). Also, it is gratifying (and not evident) that the Newton constant turns out to be positive. The mechanism is basically the same as the "gravity bag" mechanism discussed in [15]. Its 4-dimensional version is clearly more complicated and currently under investigation, however at least certain aspects of the mechanism generalize [14].
However, since the gravitational coupling is dynamical itself, the above linearized treatment of the coupling is justified only as long as the perturbations of the radial K a µν is negligible, i.e.
For the AdS 2 backgrounds under consideration, this implies that the intrinsic curvature perturbation is smaller than the background constant curvature. This is clearly inadequate for physical gravity, however the basic mechanism should extend beyond this regime for backgrounds where the extrinsic curvature dominates the intrinsic one, such as cylinders or generalizations.

Induced metric curvature
It is instructive to compute also the Ricci tensor for the induced metric g µν . Recall the decomposition of A a into tangential and normal components (102). We have Writing the metric perturbation as one finds noting that Ric µν [g] = 1 2 Ric[g]g µν and the identity (125). Therefore As a consistency check, we note that the tangential variations A µ drop out, since they correspond to a diffeomorphism. Since g = e σ G, this is related to δ A Ric[G] up to conformal rescaling contributions.

Gauge theory point of view
In this final section, we disentangle and essentially solve the model using the gauge theory point of view. Recall the decomposition (102) of A a into normal and tangential components. For the normal perturbations A ⊥ , we can use the identity The second is a scalar wave equation for A ⊥ , and χ can be seen as part of its source, determined by the first equation. For distances below the "cosmological" scales, the mass term can be neglected, leading to massless wave equations with source determined by T µν as above.
A remark on the relation with the noncommutative gauge theory point of view is in order. The usual gauge fields A µ in the gauge theory interpretation are related to our tangential perturbations as since J µa = η µa if x a for a = 0, 1 are normal embedding coordinates, cf. (84). Thus up to some constant. This is gauge invariant (more precisely it transforms as a scalar field under noncommutative gauge transformations i.e. symplectomorphisms), and encodes the only physical degree of freedom in 2D gauge theory. Similarly, A ⊥ can be interpreted as noncommutative scalar field in the noncommutative gauge theory. Therefore ∂ µ A µ and A ⊥ completely capture the physics of the system, which is described by (127) at the semi-classical (Poisson) level. It is also worth pointing out that the radial and tangential perturbations mix as observed in [14], but we were able to disentangle them in the 2dimensional case.

Conclusion
We studied the fuzzy version of 2-dimensional de Sitter and Anti-de Sitter space, and some of the associated physics. The quantization map is discussed in detail, and we obtained explicit formulae for the functions on the fuzzy hyperboloid corresponding to unitary irreducible representations of SO(2, 1). This should provide the basis for further work on the associated non-commutative field theory on a curved space-time with Minkowski signature. Moreover, we consider a matrix model which admits fuzzy (A)dS 2 as solution, and study the resulting dynamics of the geometry. This allows to study the general ideas of emergent geometry in matrix models on a simple curved background with Minkowski signature. Although the model is modified as compared with the IKKT model by adding a cubic term, it is an interesting toy model which allows to essentially solve the resulting dynamics. We find that the transversal brane perturbations indeed couple to the energy-momentum tensor as emphasized in [14], and we also find a mixing between tangential and transversal perturbations in the gauge theory point of view. The brane dynamics leads to a reasonable linearized gravity theory, related to Henneaux -Teitelboim gravity in 2 dimensions. It is remarkable that this happens through the bare matrix model action, without adding any gravity terms and without invoking any quantum effects. The mechanism does not require a strong-coupling regime. Even though the present toy model is not of direct physical relevance, it is nevertheless useful to clarify the dynamics of the branes and their geometry, as a step towards higher-dimensional more physical matrix models such as the IKKT model. It would also be interesting to study a finite-dimensional realization of the matrix model numerically, following [22]. This might serve as a toy model and testing ground for the case of Minkowski signature, as a step towards the higher-dimensional case. Therefore Finally, we have Therefore where we used due to (132). Now we can use the equations of motion (107), which give ∇ µ (J a µ G A a ) = 1 4 g Y M e σ/2 ∇ µ g µν ∇ ρT νρ = 1 4 g Y M e −σ/2 ∇ µ ∇ νT µν K a G A a = − 1 8 g Y M e 3σ/2 K a K a T = 1 2 g Y M e −σ/2 R −2 T (139) recalling that J a K a = 0, as well as Putting these together, we finally arrive at