2D fuzzy Anti-de Sitter space from matrix models
aa r X i v : . [ h e p - t h ] S e p UWThPh-2013-22
2D fuzzy Anti-de Sitter space from matrix models
Danijel Jurman ∗ , , Harold Steinacker † , ∗ Theoretical Physics Division, Rudjer Boskovic InstituteP.O. Box 180, 10002 Zagreb, Croatia † Faculty of Physics, University of ViennaBoltzmanngasse 5, A-1090 Vienna, Austria
Abstract
We study the fuzzy hyperboloids
AdS and dS as brane solutions in matrixmodels. The unitary representations of SO (2 ,
1) required for quantum fieldtheory are identified, and explicit formulae for their realization in terms offuzzy wavefunctions are given. In a second part, we study the ( A ) dS branegeometry and its dynamics, as governed by a suitable matrix model. In par-ticular, we show that trace of the energy–momentum tensor of matter inducestransversal perturbations of the brane and of the Ricci scalar. This leads to alinearized form of Henneaux–Teitelboim–type gravity, illustrating the mecha-nism of emergent gravity in matrix models. [email protected] [email protected] ontents AdS and gravity 15 There has been a great amount of work on noncommutative field theory on the the fuzzysphere and similar compact quantum spaces. Part of their appeal stems from the factthat the space of functions on these spaces has a simple group-theoretical structure and isfinite-dimensional, reflecting their finite symplectic volume. This leads to mathematicallywell-controlled toy models for noncommutative field theory and geometry, see e.g. [1–9] andreferences therein. However, most of the work so far has been for spaces with Euclideansignature, and it would be desirable to know more about fuzzy spaces with Minkowskisignature.In this paper, we study in detail 2-dimensional fuzzy de Sitter space dS and Anti-deSitter space AdS , which are quantized homogeneous spaces with Minkowski signature andnon-vanishing curvature. Fuzzy AdS has been studied previously in [10, 11]. In the firstpart of this paper, we elaborate the space of functions on these fuzzy hyperboloids, andprovide explicit formulae for the square-integrable wavefunctions corresponding to unitaryirreducible representations of SO (2 , on fuzzy AdS and For a discussion in the undeformed case see e.g. [12] and references therein. S . In particular, this also allows to establish the required quantization map for the fuzzygeometry.In a second part, we consider a matrix model which describes dynamical fuzzy AdS and dS spaces as brane solutions. As discussed in [13], this leads to a dynamical effectivegeometry on the branes, determined by a combination of the embedding geometry of thebrane and its Poisson structure. The present 2-dimensional example provides an interestingtoy model for emergent gravity, with a non-trivial curvature background. We study theperturbations around the AdS solutions, and their dynamics in the presence of matter.This is interesting because the extrinsic curvature of the brane leads to a coupling ofthe linearized matrix perturbations to the energy-momentum tensor , as pointed out in[14, 15]. More precisely, the transversal perturbations of the brane couple to the trace ofthe energy-momentum tensor of matter, due to the extrinsic curvature. It turns out that theperturbations of the effective metric are governed by a linearized Henneaux–Teitelboim–type 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 ofemergent gravity in Yang-Mills matrix models. However, this result is restricted to thelinearized regime.In 4 and higher dimensions, the dynamics of the effective geometry is complicated dueto a mixing between tangential and transversal brane perturbations [14], which prohibitsa full understanding at present. A similar mixing is observed here, but we are able todisentangle the coupled wave equations, and thereby essentially solve the perturbativedynamics. Therefore the present 2-dimensional case should serve as a useful step towardsunderstanding the more complicated higher-dimensional case. There are three types of two-dimensional non-compact spaces with constant curvature,given by the Anti-de Sitter space
AdS , de Sitter space dS and the hyperbolic orLobachevsky plane H . In this paper we discuss AdS and dS , which can be naturallyrealized as the one-sheeted hyperboloid embedded in R through x a x b η ab = − ( x ) − ( x ) + ( x ) = − R . (1)In terms of conformal coordinates − π/ < σ < π/ π < τ ≤ π , the embedding ofclassical Anti-de Sitter space AdS is given by x = R cos τ cos σ , x = R sin τ cos σ , x = R tan σ. (2) rather than just its derivative, as on trivially embedded branes. g µν = η ab ∂ µ x a ∂ ν x b , η ab = diag( − , − , , µ, ν = σ, τ , (3) g ττ = R cos σ , g σσ = − R cos σ , g στ = 0 , (4)with closed time-like circles around x = const . De Sitter space dS is obtained from AdS by switching the roles of the time and space, thus changing the overall sign in themetric. The circles x = const are then space-like, and there are no closed time-like curves.Both AdS and dS admit the group SO (2 ,
1) or its cover SU (1 ,
1) as isometries, gen-erated by vector fields K a , a = 1 , , K = − i cos τ sin σ∂ τ − i sin τ cos σ∂ σ , K = − i sin τ sin σ∂ τ + i cos τ cos σ∂ σ , K = − i∂ τ , (5)which close su (1 ,
1) Lie algebra with respect to commutators[ K a , K b ] = if abc K c (6)or explicitly [ K , K ] = − iK , [ K , K ] = iK , [ K , K ] = iK . (7)The Casimir operator of su (1 ,
1) Lie algebra is defined as C = − ( K ) − ( K ) + ( K ) . (8)As usual, it is convenient to introduce the ladder operators K ± = K ± iK , (9)which satisfy the commutation relations (cid:2) K , K ± (cid:3) = ± K ± , (cid:2) K + , K − (cid:3) = − K . (10)Then unitary irreducible representations of SO (2 ,
1) are spanned by a basis | j, m i of weightstates, where j is related to the eigenvalue of the Casimir C , and m is the eigenvalue of K and the action of K ± on | j, m i produces a state with weight m ± K K ± | j, m i = ( m ± K ± | j, m i ∼ | j, m ± i . (11)A chain of states obtained by the successive action of K − operator terminates if there existstate such that K − | j, m i = 0 . (12)Denoting this lowest weight by j = m , it follows that0 = K + K − | j, j i = (cid:0) − C + K ( K − (cid:1) | j, j i ⇒ C = j ( j −
1) (13) They can be avoided by passing to the universal cover of
AdS . | j, j i of lowest weight, via | j, j + m i ∼ K + m | j, j i . (14)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 conditionimplies C <
0, and the states belong to the unitary irreducible continuous representations.In general, the resulting structure of irreducible representations is as follows: K | j, m i = m | j, m i ,K + | j, m i = a m +1 | j, m + 1 i ,K − | j, m i = a m | j, m − i , (15)where a m = p m ( m − − j ( j − . (16)The finite-dimensional irreducible representations of SU (1 ,
1) are obtained for j ∈ − N / | j | representations V | j | of SU (2) with C = −| j | ( | j | + 1). All unitary irreducible representations are infinite-dimensional, and fallinto one of the following classes [17]: • The discrete series of the highest and the lowest weight representations D + j , j ∈ N > : H j = {| j, m i ; m = j, j + 1 , · · · ; m ∈ N } ,D − j , j ∈ N > : H j = {| j, m i ; m = − j, − j − , · · · ; − m ∈ N } , (17)characterized by C = j ( j − ≥ • The principal continuous series P s , s ∈ R , < s < ∞ , j = 12 + is, H j = {| j, m i ; m = 0 , ± , ... ; m ∈ Z } (18)labeled by a real number s and C = − (cid:0) s + (cid:1) < − / • The complementary series P cj , / < j < , j ∈ R , H j = {| j, m i ; m = 0 , ± , ... ; m ∈ Z } (19)with − / < C < In order to carry out the quantization of ( A ) dS , it is useful to organize the space offunctions on ( A ) dS in terms of irreducible representations of SU (1 , (cid:3) g , (cid:3) g = 1 p | g | ∂ µ p | g | g µν ∂ ν = cos σR (cid:0) ∂ τ − ∂ σ (cid:1) = K + K − K R , (20) We only consider representations with integer weights for simplicity. su (1 , g = det( g µν ), and g µν is inverseof the metric. We can thus decompose any function on the hyperboloid into eigenfunctionsof (cid:3) g , (cid:3) g φ + αφ = 0 , (21)and label the solutions by j and m as above. The solutions corresponding to the finite-dimensional representations are realized by polynomial functions Pol(x a ); they are of coursenot normalizable on ( A ) dS . The square-integrable functions corresponding to unitaryirreducible representations are given explicitly in terms of hyper-geometric functions φ jm = e − imτ cos j σ (cid:20) a F (cid:18) j + m , j − m ,
12 ;sin σ (cid:19) + b sin σ F (cid:18) j + m +12 , j − m +12 ,
32 ;sin σ (cid:19)(cid:21) , (22)where C = j ( j −
1) = R α (23)is the Casimir. For AdS , the scalar fields corresponding to positive or negative energyunitary representations belong to the discrete representation D ± j , with α >
0. Then theequations for the lowest weight state have a unique solution K φ jj = jφ jj K − φ jj = 0 (cid:27) ⇒ φ jj ∼ e ijτ cos j σ (24)and the spectrum is non-degenerate. On the other hand the states given by (22) with α < dS . Putting these together, we have thefollowing decomposition of functions on the hyperboloid ( A ) dS L (( A ) dS ) = ⊕ J ≥ D + J ⊕ J ≥ D − J ⊕ Z ∞ dSP S (25)along with the space of polynomial functions Pol(x a ).In the following we discuss fuzzy versions of these non-compact spaces, and their asso-ciated spaces of functions. As a starting point, we note that the natural SO (2 , ω = Rκ cos σ dτ ∧ dσ (26)with dω = 0, introducing a scale perameter κ . Its inverse defines the Poisson bracket oftwo functions { f, g } = κ cos σR ( ∂ τ f ∂ σ g − ∂ σ f ∂ τ g ) . (27)We can now look for a quantization of this Poisson manifold M , cf. [20]. This means thatthe 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,5he group-theoretical structure of ( A ) dS provides a natural and explicit quantization, inanalogy to the case of the fuzzy sphere [1]. As a first step, we note that the Poisson bracketsof the embedding functions x a satisfy the Lie algebra of SO (2 , { x a , x b } = κf abc x c (28)where f abc are structure constants of SO (2 , SO (2 ,
1) vector fields (5) are given by K a = iκ { x a , . } . (29) In analogy to the fuzzy sphere [1], we define fuzzy two-dimensional hyperboloid in termsof three hermitian matrices (or operators) X a , which are interpreted as quantization of theembedding functions x a . In view of (28), we impose the following relations[ X a , X b ] = iκ f abc X c , (30)where f abc are structure constants of the Lie algebra su (1 , X a are rescaled su (1 ,
1) generators, and we assume that they act on a certain irreducible unitary represen-tation H j of the Lie algebra. We can then write the Casimir operator as X a X b η ab = κ j ( j − . (31)Since H j is assumed to be irreducible, the X a generate the full algebra A of operators on H j A := End ( H j ) ∼ = H j ⊗ H ∗ j , (32)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 : K a ⊲ Φ = 1 κ [ X a , Φ] , Φ ∈ A . (33)We now specify the representation H j . Since the matrices X a should be interpreted asquantized embedding functions x a of the hyperboloid and comparing the spectrum of X with the range of x ∈ h−∞ , ∞i , we choose H j to be a principal continuous representation ,in accord with [10].We can furthermore define an invariant scalar product(Φ , Φ ) = TrΦ † Φ , Φ , Φ ∈ A . (34) The complementary representation is rejected because it does not admit a semi-classical limit for fixedcurvature, as explained in section 3.2. 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 de-composing A = H j ⊗ H ∗ j into irreducible unitary representations of su (1 , | J M i whichbelong to a particular unitary irreducible representation in H j ⊗ H j are given by | J M i = X m ,m C j j Jm m M | j m i ⊗ | j m i . (35)Here the C ’s are the Wigner coefficients, which vanish unless M = m + m . In the specialcase of H j ⊗ H ∗ j , we represent the state (35) as a matrix Φ JM Φ JM = X m m D j j Jm m M | jm ih jm | , (36)where the D ’s vanish unless M = m − m ∈ Z . Since we chose the principal continuousrepresentation H j ∼ = P s , one obtains the following decomposition of the space of functions A into unitary modes [18]: P s ⊗ P ′ s = ⊕ J ≥ D + J ⊕ J ≥ D − J ⊕ Z ∞ dSP S , (37)along with the space of polynomial functions Pol(X a ). In the next section we will recoverthis result and obtain the corresponding fuzzy wavefunctions explicitly, which solve theeigenvalue equations η ab K a ⊲ K b ⊲ Φ JM = − κ (cid:3) Φ JM = η ab κ [ X a , [ X b , Φ JM ]] = J ( J − JM ,K ⊲ Φ JM = 1 κ [ X , Φ JM ] = M Φ JM . (38) We can determine the fuzzy wavefunctions Φ JM explicitly, using their definition as irreduciblerepresentations of SO (2 , A , the matrix Φ JM actson | jn i ∈ H j asΦ JM | jn i = X m D j j Jmm − MM | jm ih jm − M | jn i = D j j Jn + M nM | jn + M i . (39)Defining the matrix D JM ( K ) by its action on | jn i D JM ( K ) | jn i = D j j Jnn − MM | jn i (40)7nd using defining property Γ( x + 1) = x Γ( x ) of Gamma function, we can express Φ JM forinteger M using (15) asΦ JM = D JM ( K ) s Γ( K − M − j + 1)Γ( K − M + j )Γ( K − j + 1)Γ( K + j ) K + M == D JM ( K ) p ( K − j )( K + j − K + ! M , M > JM = K − M s Γ( K − M − j + 1)Γ( K − M + j )Γ( K − j + 1)Γ( K + j ) D JM ( K ) == K − p ( K − j )( K + j − ! M D JM ( K ) † , M < . (42)To derive the final expressions (41) and (42) one applies the identity s Γ( K − M − j + 1)Γ( K − M + j )Γ( K − j + 1)Γ( K + j ) K + M = p ( K − j )( K + j − K + ! M , (43)which can be verified using K + F ( K ) = F ( K − K + , (44)which follows from (10). We note that the expressions (41) and (42) are hermitian conju-gates of each other. This reflects the fact that Φ JM † is a solution of (38) with eigenvalue − M if Φ JM is a solution with eigenvalue M .The above considerations apply to any representation. For the discrete series represen-tations D + J in (37) with J being integer, the basis of states is completely determined bythe minimal weight state annihilated by K − . Acting with K − on (41) we obtain[ K − , Φ JM ] = p ( M − J )( M + J − JM − == h D JM ( K + 1) p ( K + j )( K − j + 1) − D JM ( K ) p ( K − M + j )( K − M − j + 1) i ×× p ( K − j )( K + j − K + ! M − . (45)Specializing this to the case M = J , we see that the expression in square bracket mustvanish D JJ ( K + 1) s Γ( K + j + 1)Γ( K − j + 2)Γ( K − J − j + 2)Γ( K − J + j + 1) −− D JJ ( K ) s Γ( K + j )Γ( K − j + 1)Γ( K − J + j )Γ( K − J − j + 1) = 0 . (46)8ince K takes only integer values here, we can conclude that D JJ ( K ) = s Γ( K − J + j )Γ( K − J − j + 1)Γ( K + j )Γ( K − j + 1) (47)(up to normalization). Finally, for the lowest weight state in D + J using (43) we obtainΦ JJ = (cid:18) K − j )( K + j − K + (cid:19) J (48)in agreement with findings of [10,11]. The highest weight state in D − J is given by hermitianconjugate of (48).For the principal continuous representation P S with J = 1 / iS in (37), the matrices D JM are solutions of second order difference equation obtained from (45) applying K + onit: (cid:2) ( M − J )( M + J − − ( K − j )( K + j − − ( M − K − j )( M − K + j − (cid:3) C JM ( K ) = p ( K − j )( K + j − M − K + j )( M − K − j + 1) C JM ( K −
1) + p ( K + j )( K − j + 1)( M − K − j )( M − K + j − C JM ( K + 1) . (49) Here we find it convenient to write D JM as D JM ( K ) = C JM ( K ) e iπK , (50)where C JM ( K ) is a matrix with elements given by the Wigner coefficients for the principalcontinuous representations of su (1 ,
1) in (37). This can be seen after noting that this secondorder difference equation is a special case of equation for the general Wigner coefficients asfound in [19]. Finally, we can express C JM in terms of two independent solutions C JM ( K ) = s Γ( M − K − j + 1)Γ( M − K + j )Γ( K + j )Γ( K − j + 1) aG JM ( K ) + bG − JM ( K )Γ( M − K + J − j + 1)Γ( M − K − J − j − . (51) where G JM ( K ) is the hypergeometric series G JM ( K ) = F ( J, j + J − , J − M ; K + J + j − M, J ) , (52)defined by F ( a, b, c ; d, e ) = ∞ X k =0 Γ( a + k )Γ( b + k )Γ( c + k )Γ( d )Γ( e ) k !Γ( a )Γ( b )Γ( c )Γ( d + k )Γ( e + k ) . (53)To summarize, we have obtained explicit matrices Φ JM of the formΦ JM = (cid:26) F JM ( X ) X + M , M ≥ F JM ( X ) X − M , M ≤ , (54) The solutions are degenerate in the case of J being integer. A = H j ⊗ H ∗ j into unitary representations of SO (2 , φ JM = (cid:26) f JM ( x ) x + M , M ≥ f JM ( x ) x − M , M ≤ x + = x + ix on the hyperboloid. Due to the relation with the Casimir, the spectrum ofmatrix d’Alembertian κ (cid:3) in (38) and the classical d’Alembertian (20) coincide. Includingalso the space of polynomial functions Pol( X a ), this is the basis for interpreting the matrixalgebra A as quantized algebra of functions over hyperboloid. Now consider the classical hyperboloid M as a Poisson manifold equipped with the Pois-son structure (27). The quantization of such a Poisson manifold is defined in terms of aquantization map Q , which is an isomorphism of vector spaces from the space of functions C ( M ) on M to some (operator) algebra AQ : C ( M ) → A , f ( x )
7→ Q ( f ( x )) (56)which is compatible with the Poisson structure { f, g } = θ µν ∂ µ f ∂ ν g , satisfying Q ( f g ) − Q ( f ) Q ( g ) → θ (cid:16) Q ( i { f, g } ) − [ Q ( f ) , Q ( g )] (cid:17) → θ → . (58)Clearly Q ≡ Q θ depends on the Poisson structure θ , and the limit θ → Q is anisomorphism of vector spaces , one can then define the semi-classical limit of some fuzzywavefunction F ∈ A as the inverse f = Q − ( F ) of the quantization map. This is consistentas θ →
0, provided commutators are replaced by the appropriate Poisson brackets, andhigher order terms in θ are neglected.In general, there is no unique way of defining Q . However in the case at hand, thereis a natural definition of Q , based on the decomposition of C ( M ) and A into irreduciblerepresentations of SO (2 , JM and φ JM of A resp. C ( M ) as obtained above, we define Q ( φ JM ) = Φ JM , (59)so that Q is an isometry for the unitary representations. This can be extended to the poly-nomials Pol( x a ), corresponding to finite-dimensional non-unitary irreducible representa-tions of SO (2 , Q (Pol( x a )) Sometimes one only requires Q to be an isomorphism only on the space of functions with momentabelow some UV cutoff. X a , a = 1 , , x a , a = 1 , ,
3, we define Q ( x a ) = X a . (60)Comparing the embedding equation x a x a = − R with the Casimir constraint X a X a = κ j ( j −
1) (31), we are led to impose κ (cid:18) s + 14 (cid:19) = R = const, (61)using j = 1 / is for the principal continuous representation H j . Therefore the semi-classical limit κ → s → ∞ . This is the analog of N = dim H → ∞ in the caseof fuzzy sphere.To establish the required properties of Q , we recall that all fuzzy wavefunctions canbe written in the following “normal form” (54)Φ JM = F JM ( X ) X + M , φ JM = f JM ( x ) x + M , M ≥ M <
0. We claim thatlim κ → F JM = f JM (63)as functions in one variable. To see this, observe that in the limit κ → κ → κ [ X ± , F ( X )] = ∓ F ′ ( X ) X ± , (64)lim κ → κ [ X − , X + M ] = 2 M X X + M − , (65)as a consequence of the Lie algebra relations. In the classical case, the correspondingrelations are i { X ± , f ( x ) } = ∓ f ′ ( x ) x ± , (66) i { x − , x + M } = 2 M x x + M − . (67)Therefore the action of the matrix Laplacian (38) on Φ JM in the limit κ → lim κ → κ (cid:3) Φ JM = − h(cid:16) X + R (cid:17) F ′′ JM ( X ) + 2( M + 1) X F ′ JM ( X ) + M ( M + 1) F JM ( X ) i X + M (68) has precisely the same form as the action of the classical Laplacian R (cid:3) g φ JM = − κ { x + , { x − , φ JM }} + 1 κ { x , { x , φ JM }} + iκ { x , φ JM } = − J ( J − φ JM . (69) this is also the reason why the complementary series has been rejected for H j . Our aim is to establish and clarify the required properties, without claiming mathematical rigor. Q in such a way that Q ( f JM ( x ) x + M ) → f JM ( X ) X + M as κ →
0. In particular, this provides the appropriatedefinition of Q for the principal continuous representation P S (which is doubly degener-ate), 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 inthe limit κ →
0. Consider the product of two matrix modes as above, expanded up toleading order in κ Q (Φ JM ) Q (Φ J ′ M ′ ) = Φ JM Φ J ′ M ′ = F JM X + M F J ′ M ′ X + M ′ = F JM [ X + M , F J ′ M ′ ] X + M ′ + F JM F J ′ M ′ X + M + M ′ = ( F JM F J ′ M ′ − M κF JM F ′ J ′ M ′ + o ( κ )) X + M + M ′ , (70)for M, M ′ ≥
0. Then (57) follows immediately using (63). Subtracting the same com-putation with the factors reversed, (58) follows. A similar computation applies to modeswith mixed or negative M . Therefore Q is indeed a quantization map for our Poissonstructure. Using the decomposition of A into the above modes, analogous statements canbe made for the de-quantization map Q − , which provides the semi-classical limit of thefuzzy wavefunctions.Finally, consider the trace of some normalizable wavefunctions with weight M = 0,2 π TrΦ J † Φ J = 2 π ∞ X m = −∞ F J ∗ ( κm ) F J ( κm ) κ → → πκ − Z dx f J ∗ ( x ) f J ( x )= Z ω φ ∗ J φ J , (71)using the explicit form of H j = P s (18), where ω is the symplectic form (26). This compu-tation is easily generalized to show that2 π Tr Q ( f ) Q ( g ) κ → → Z ω f g (72)as long as the integrals are bounded. This is guaranteed for the spaces of unitary wave-functions discussed above.To summarize, in the semiclassical limit ∼ defined as de-quantization map expandedup to leading order in κ , we can use the following relationsΦ JM ∼ φ JM , X a ∼ x a , [ F, G ] ∼ i {Q − ( F ) , Q − ( G ) } , [ X a , ] ∼ i { x a , } , π Tr ∼ Z ω (73)which we use in the following sections. Consider now three hermitian matrices X a = ( X a ) † ∈ M at ( ∞ , C ) for a = 1 , ,
3, whichtransform in the basic 3-dimensional representation of SO (2 , SO (2 ,
1) symmetry as well astranslations X a → X a + c a
1l has the form S [ X ] = − πg Y M Tr (cid:16) [ X a , X b ][ X a , X b ] + ig f abc [ X a , X b ] X c (cid:17) (74)for suitable constants, where embedding indices are raised and lowered with η ab . Thematrices X a are understood to have dimension length, and accordingly [ g Y M ] = L . Thismodel is invariant under SO (2 ,
1) rotations, translations as well as gauge transformations X a → U X a U − for unitary operators U . The equations of motion are obtained as4 (cid:3) X a = 3 ig f abc [ X b , X c ] , (cid:3) ≡ [ X a , [ X a , . ]] . (75)Now consider the ansatz X a = κK a , a = 1 , , SO (2 , X a , X b ] = iκ f abc X c ,X a X a = κ C | H = − κ ( s + 14 )1l H = − R H , (cid:3) X a = κ C | ad X a = 2 κ X a (77)where C is the quadratic Casimir of SO (2 , κ, R are positive numbers. As discussedbefore, we take H = H j to be the principal continuous series representation, so that X a ∈ End ( H ) and R κ = ( s + 14 ) = − C | H . (78)Thus the equations of motion (75) are solved by this ansatz provided4 κ + 3 κg = 0 . (79)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 thesemi-classical limit, following [13]. Recall that the matrices X a are interpreted as quantizedCartesian embedding functions X a ∼ x a : M ֒ → R . (80) Of course the matrix model action is divergent on this background, however this is not a problem. Weonly need to require that the perturbations lead to a finite variation of the action. This could be takeninto account by subtracting certain background terms, which we do not write down for brevity. M is given by g µν = ∂ µ x a ∂ ν x a . (81)For the hyperboloid solutions under consideration x a x a = − R holds, so that the inducedmetric is that of AdS . However, we are interested in the effective metric which governsphysical fields in the matrix model. To identify the effective metric in the semi-classicallimit, we note that the kinetic term (with two derivatives) for e.g. scalar fields Φ in thematrix model arises from an action of the form S [ φ ] = − π g Y M
Tr [ X a , Φ][ X a , Φ] ∼ g Y M Z ω θ µµ ′ θ νν ′ g µ ′ ν ′ ∂ µ φ∂ ν φ = − g Y M Z d x p | G | e − σ/ G µν ∂ µ φ∂ ν φ = − Z d x p | G | e − σ/ G µν ∂ µ ϕ∂ ν ϕ, (82)using the semi-classical correspondence rules (73). Here the scalar fields are made dimen-sionless via Φ ∼ φ = g / Y M ϕ , and ω = 12 θ − µν dx µ dx ν ,G µν = − g − Y M θ µµ ′ θ νν ′ g µ ′ ν ′ = e − σ g µν ,e − σ = g − Y M | det θ µν || det g µν | . (83)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 signis dictated by the action (74) resp. (82), which must have the form S = R dt ( T − V ). Forthe action (74) it means that the effective metric is indeed that of AdS , while fuzzy dS can be obtained by changing the overall sign of the action. This choice of signs is possibleonly 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, dueto the Weyl symmetry G µν → e α G µν . Here we choose (83) for simplicity; our main goal isto illustrate how such an effective metric responds to matter perturbations in the presentmatrix model.The relation G ∼ g is particular for 2 dimensions, and can be seen in coordinates where g µν = diag( − ,
1) at some given point p ∈ M . Consider the point p N = ( R, ,
0) in thehomogeneous
AdS space. Its tangent space is parallel to the ( x x ) plane, so that wecan use x µ = ( x , x ) as local coordinates. In these “normal embedding” coordinates wehave g µν = diag( − ,
1) at p N , and θ = { x , x } = κf R = κR . On the other hand θ µν = { x µ , x ν } = g Y M e − σ/ ǫ µν using (83), and we obtain e − σ/ = | g − Y M x | = g − Y M κR. (84) For example, the radial components of the non-abelian fluctuations on a stack of coincident branesrealize such scalar fields. A detailed analysis of general abelian perturbations will be given below.
14e note that the matrix Laplace operator (77) for the unperturbed hyperboloid backgroundis related to the geometric Laplace operator in the semi-classical limit (cid:3) Φ ∼ −{ x a , { x a , φ }} = g Y M √ G ∂ µ ( √ GG µν ∂ ν φ ) = g Y M (cid:3) G φ . (85)Finally, it is easy to add fermions the matrix model, via the action S [ ψ ] = Ψ Γ a [ X a , Ψ] + m ψ ΨΨ . (86)Here Ψ = (cid:18) ψ ψ (cid:19) , ψ α ∈ A (87)is a 2-component spinors of SO (2 , a satisfy the Clifford algebra of SO (2 , a Γ b + Γ b Γ a = 2 η ab . (88) AdS and gravity We introduce some useful geometrical structures which apply to general M ⊂ R . The“translational currents” J aµ = ∂ µ x a (89)span the tangent space of M ⊂ R , while K aµν = ∇ [ g ] µ J aν = K aνµ (90)characterizes the extrinsic curvature and is normal to the brane with respect to the em-bedding metric, J aµ K aνη = 0 . (91)In particular, K a = K aµν G µν = (cid:3) G x a (92)is a normal vector to M ⊂ R . For the present AdS solution, one can easily computethe curvature of the connection ∇ [ G ] = ∇ [ g ] ≡ ∇ , K aµν = R − g µν x a = 12 G µν K a ,K a = 2 e − σ R − x a = − Ric[ G ] x a (93) 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 Laplacianonly for the unperturbed backgrounds, where (85) is sufficient. In general, this holds for (cid:3) g rather than (cid:3) G , but in the 2-dimensional case both statements are true. (cid:3) g x a = R x a on AdS . The Riemann curvature tensor can beobtained e.g. from the Gauss-Codazzi theorem, and is given byR µνηρ = − R − ( g µη δ ρν − g νη δ ρµ ) , Ric µν = 12 Ric[ g ] g µν = 12 Ric[ G ] G µν , Ric[ g ] = − R − , Ric[ G ] = − R − e − σ (94)using (84), and recalling x a x a = − R . Using the above relations along with (77), theembedding functions x a satisfy ( (cid:3) G + Ric[ G ]) x a = 0 . (95)Now consider small fluctuations around the solutions ¯ X a of the above matrix model,parametrized as X a = ¯ X a + A a ( ¯ X ) . (96)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 fuzzysphere [6]. Then the radial component can be interpreted as scalar field on M , and thetangential components in terms of (noncommutative) gauge fields. This interpretation iscertainly appropriate for the non-abelian components, which arise on a stack of n coincidingsuch branes. However since the trace- U (1) components change the effective metric G µν on M , it is more natural to interpret them in terms of geometrical or gravitational degrees offreedom; note that there is no charged object under this U (1). In this section we elaboratesome aspects of the resulting 2-dimensional effective or ”emergent“ gravity .In the semi-classical limit, the matrix model action expanded to second order in A a around the basic AdS solution is given by S [ X ] ∼ g Y M Z ω (cid:16) { x a , A b }{ x a , A b } + { x a , A b }{A a , x b } + ( κ + 3 g f abc {A a , A b } x c (cid:17) = 2 g Y M Z ω (cid:16) g Y M A b (cid:3) G A b − f + (2 κ + 3 g f abc x c {A a , A b } (cid:17) = 2 g Y M Z ω (cid:16) g Y M A b (cid:3) G A b − f (cid:17) (97)dropping the linear as well as the f abc term which vanish due to the equations of motion(79), and using Z { x a , A b }{A a , x b } = − Z A b { x a , {A a , x b }} = Z A b ( {A a , { x b , x a }} + { x b , { x a , A a }}} = Z ( κf abc {A a , A b } x c + {A b , x b } , { x a , A a } ) . (98) The word ”emergent“ indicates that the metric arises from other, more fundamental degrees of freedom. f = {A a , x a } (99)can be viewed as gauge fixing function, since it transforms as f → f + { x a , { x a , Λ }} = f − g Y M (cid:3) G Λ (100)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 allfields on M couple to the effective metric G µν , so that the metric perturbations coupleto matter via the energy-momentum tensor. The linearized metric fluctuation is given by δ A g µν = J aµ ∂ ν A a + J aν ∂ µ A a = ∇ µ A ν + ∇ ν A µ − K aµν A a (101)where we decompose the perturbations into tangential and transversal ones A ⊥ = K a A a , A µ = J aµ A a . (102)Using 2 K aµν = e σ g µν K a (93), the perturbation of the effective metric in Darboux coordinatescan be written as δ A G µν = − g − Y M θ µµ ′ θ νν ′ δ A g µν = − g − Y M θ µµ ′ θ νν ′ ( ∇ µ A ν + ∇ ν A µ ) − e σ G µν K a A a . (103)Therefore δ A S M = − Z d x √ G T µν δ A G µν = Z d x √ G ( e σ T K a − ∇ µ ˜ T µν J aν ) A a (104)noting that ∇ θ µν = 0, where T = T µν G µν , and we define˜ T µν = g − Y M θ νν ′ θ µµ ′ T µ ′ ν ′ (105)for convenience. Thus the normal component A ⊥ couples to the trace of the energy-momentum tensor, while the tangential components couple to its derivative. This illustratesthe observation [14] that a non-derivative coupling of the embedding perturbations to theenergy-momentum tensor arises on branes with extrinsic curvature. Using the on-shellcondition (79) for the background and q | G µν || θ µν | = g Y M e σ/ , (106)we obtain the semi-classical equations of motion (cid:3) G A a = 18 g Y M e σ/ (cid:0) − e σ T K a + 2 J aµ ∇ ν ˜ T µν (cid:1) . (107)Note that the normal component couples to T via the extrinsic curvature. This is thecrucial ingredient for gravity, as we will see below. We ignore possibly different conformal factors for different types of matter, for the sake of illustratingthe mechanism in a simple toy model. .1 Curvature perturbations and gravity Now we can obtain the curvature perturbations induced by matter. Since in 2 dimensionsRic µν [ G ] = G µν Ric[ G ] where Ric[ G ] is the Ricci scalar, we will restrict ourselves to studythe linearized perturbations of Ric[ G ]. This can be computed using δR µν [ G ] = − ∇ µ ∂ ν ( G ρη δG ρη ) − (cid:3) G δG µν + ∇ ( µ ∇ η δG ν ) η , (108)which implies δ A Ric[ G ] = ( (cid:3) G + 12 Ric[ G ])( G µν δ A G µν ) − ∇ µ ∇ ν δ A G µν . (109)The perturbation of the effective metric in Darboux coordinates can be written as follows(cf. (103)) δ A G µν = − g − Y M θ µµ ′ θ νν ′ ( J aµ ′ ∂ ν ′ A a + J aν ′ ∂ µ ′ A a ) (110)using δθ µν = 0. After some computations given in the appendix, the corresponding pertur-bation of the Ricci tensor is obtained as δ A Ric[ G ] = 12 g Y M e σ/ (cid:16) R − T + ∇ µ ∇ ν ˜ T µν (cid:17) = 8 πG N (cid:0) T + R ∇ µ ∇ ν ˜ T µν (cid:1) , πG N = e σ/ g Y M R = κe σ R . (111)This can be seen as linearization of the following gravity modelRic[ G ] − Λ = 8 πG N T + O ( ∂∂T ) , Λ = − e − σ R − , (112)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 thederivative term is of order ∇∇ ˜ T ∼ e σ ∇∇ T (113)using (106), and can be neglected provided e σ ≪
1, which is compatible with G N ≪ AdS background, the result should equally apply to the dS 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 havesimply elaborated the matrix model dynamics from a geometrical point of view. The crucialcoupling to T µν arises due to the extrinsic curvature of the brane encoded in ∇ µ J aν = K aµν ,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 same18s the ”gravity bag“ mechanism discussed in [15]. Its 4-dimensional version is clearlymore complicated and currently under investigation, however at least certain aspects of themechanism generalize [14].However, since the gravitational coupling is dynamical itself, the above linearized treat-ment of the coupling is justified only as long as the perturbations of the radial K aµν isnegligible, i.e. δK aµν ≪ K aµν . (114)For the AdS backgrounds under consideration, this implies that the intrinsic curvatureperturbation is smaller than the background constant curvature. This is clearly inadequatefor physical gravity, however the basic mechanism should extend beyond this regime forbackgrounds where the extrinsic curvature dominates the intrinsic one, such as cylindersor generalizations. It is instructive to compute also the Ricci tensor for the induced metric g µν . Recall thedecomposition of A a into tangential and normal components (102). We have δ A Ric[ g ] = − ( (cid:3) g + 12 Ric[ g ])( g µν δ A g µν ) + ∇ µ ∇ ν δ A g µν . (115)Writing the metric perturbation as δ A g µν = ∇ µ A ν + ∇ ν A µ − R − g µν x a A a , (116)one finds ∇ µ ∇ ν δ A g µν = ∇ µ ∇ ν ( ∇ µ A ν + ∇ ν A µ − R − g µν x a A a )= 2 (cid:3) g ( ∇ ν A ν ) + Ric[ g ]( ∇ ν A ν ) − R − (cid:3) g ( x a A a ) , (117)noting that Ric µν [ g ] = Ric[ g ] g µν and the identity (125). Therefore δ A Ric[ g ] = − (2 (cid:3) g + Ric[ g ])( ∇ ν A ν − R − x a A a )+ 2 (cid:3) g ( ∇ ν A ν ) + Ric[ g ]( ∇ ν A ν ) − R − (cid:3) g ( x a A a )= 2 R − ( (cid:3) g + Ric[ g ])( x a A a ) . (118)As a consistency check, we note that the tangential variations A µ drop out, since theycorrespond to a diffeomorphism. Since g = e σ G , this is related to δ A Ric[ G ] up to conformalrescaling contributions. 19 .3 Gauge theory point of view In this final section, we disentangle and essentially solve the model using the gauge theorypoint of view. Recall the decomposition (102) of A a into normal and tangential components.For the normal perturbations A ⊥ , we can use the identity (cid:3) G A ⊥ = − Ric[ G ] ( K a A a ) + 2 ∇ µ K a ∂ µ A a + K a (cid:3) G A a = Ric[ G ] A ⊥ − ∇ µ A µ ) + K a (cid:3) G A a , (119)so that using the equation of motion (139) gives (cid:3) G A ⊥ = Ric A ⊥ − ∇ µ A µ ) + 12 κ R − T . (120)Similarly, consider the divergence of the tangential perturbations ∇ µ A µ = ∇ µ ( J aµ A a ) = K a A a + J aµ ∇ µ A a . (121)The tangential components of the equation of motion give J aµ (cid:3) G A a = 14 g Y M e σ/ g ηµ ∇ ν ˜ T ην , (122)so that (cid:3) G A µ = 2 ∇ ρ J aµ ∂ ρ A a + (cid:3) G J aµ A a + 14 g Y M e σ/ g ηµ ∇ ν ˜ T ην = 2 K aρµ ∂ ρ A a −
12 Ric[ G ] A µ + 14 g Y M e − σ/ G ηµ ∇ ν ˜ T ην (123)using ∇ µ K aµν = (cid:3) G J aµ = − Ric[ G ] J aµ and K aρµ = G ρµ K a . This gives ∇ µ (cid:3) G A µ = − Ric[ G ] J aρ ∂ ρ A a + K a (cid:3) G A a −
12 Ric[ G ] ∇ µ A µ + 14 g Y M e − σ/ ∇ µ ∇ ν ˜ T ην = Ric[ G ] K a A a −
32 Ric[ G ] ∇ µ A µ + 14 κR (cid:0) R − T + ∇ µ ∇ ν ˜ T µν (cid:1) . (124)Together with (cid:3) G ( ∇ µ A µ ) = −
12 Ric[G] ∇ µ A µ + ∇ µ (cid:3) G A µ , (125)it follows that the scalar field ∇ µ A µ satisfies the wave equation (cid:3) G ( ∇ µ A µ ) = − G ] ( ∇ µ A µ ) + Ric[ G ] A ⊥ + 14 κR (cid:0) R − T + ∇ µ ∇ ν ˜ T µν (cid:1) . (126)Together with (120) we we obtain the following ”almost-decoupled“ wave equations (cid:3) G χ = κR ∇ µ ∇ ν ˜ T µν , ( (cid:3) G + Ric[ G ]) A ⊥ = − G ] χ + 12 κ R − T , (127)20here χ := ∇ µ A µ − A ⊥ = J aµ ∂ µ A a . (128)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 massterm 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 inorder. The usual gauge fields A µ in the gauge theory interpretation are related to ourtangential perturbations as θ µν A ν = η µν A ν , (129)since J µa = η µa if x a for a = 0 , ∂ µ A µ ∼ θ µν ∂ µ A ν = 12 θ µν F µν (130)up to some constant. This is gauge invariant (more precisely it transforms as a scalarfield under noncommutative gauge transformations i.e. symplectomorphisms), and encodesthe only physical degree of freedom in 2D gauge theory. Similarly, A ⊥ can be interpretedas 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 thesemi-classical (Poisson) level. It is also worth pointing out that the radial and tangentialperturbations mix as observed in [14], but we were able to disentangle them in the 2-dimensional case. We studied the fuzzy version of 2-dimensional de Sitter and Anti-de Sitter space, and someof the associated physics. The quantization map is discussed in detail, and we obtainedexplicit formulae for the functions on the fuzzy hyperboloid corresponding to unitary irre-ducible representations of SO (2 , A ) dS as solution, and studythe resulting dynamics of the geometry. This allows to study the general ideas of emergentgeometry in matrix models on a simple curved background with Minkowski signature. Al-though 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. Wefind that the transversal brane perturbations indeed couple to the energy-momentum ten-sor as emphasized in [14], and we also find a mixing between tangential and transversalperturbations in the gauge theory point of view. The brane dynamics leads to a reasonablelinearized gravity theory, related to Henneaux – Teitelboim gravity in 2 dimensions. It is21emarkable that this happens through the bare matrix model action, without adding anygravity terms and without invoking any quantum effects. The mechanism does not requirea strong-coupling regime. Even though the present toy model is not of direct physical rele-vance, it is nevertheless useful to clarify the dynamics of the branes and their geometry, asa 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 modelnumerically, following [22]. This might serve as a toy model and testing ground for the caseof Minkowski signature, as a step towards the higher-dimensional case. Acknowledgments.
The work of H.S. is supported by the Austrian Fonds f¨ur Wis-senschaft und Forschung under grant P24713, and the work of D.J. was supported by thePostDoc program of the Croatian Science Fundation. This collaboration was also supportedby the Austrian-Croatian WTZ project HR 22/2012 of the OEAD.
We note the following identities J aµ J aν = g µν , J aµ K aνη = 0 (131)as well as ∇ µ J aµ = K a , (cid:3) G J aµ = ∂ µ (cid:3) G x a + Ric µν J aν = −
12 Ric[ G ] J aµ (132)which follows from (95). Then g − Y M θ µµ ′ θ νν ′ ∇ µ ∇ ν ( J aν ′ ∂ µ ′ A a ) = g − Y M θ µµ ′ θ νν ′ ∇ µ ( K aνν ′ ∂ µ ′ A a + J aν ′ ∇ ν ∂ µ ′ A a )= g − Y M θ µµ ′ θ νν ′ ( K aµν ′ ∇ ν ∂ µ ′ A a + J aν ′ ∇ µ ∂ µ ′ ∇ ν A a )= e σ K a (cid:3) G A a + 12 g − Y M θ µµ ′ θ νν ′ J aν ′ R µµ ′ ; νρ ∂ ρ A a = e σ K a (cid:3) G A a − g − Y M R − θ µµ ′ θ νν ′ J aν ′ ( g µν δ ρµ ′ − g µ ′ ν δ ρµ ) ∂ ρ A a = e σ K a (cid:3) G A a + R − J aν ∂ ν A a (133)using (94), and noting that ∇ [ g ] = ∇ [ G ] here. Similarly, we obtain using (89) g − Y M θ µµ ′ θ νν ′ ∇ µ ∇ ν ( J aµ ′ ∂ ν ′ A a ) = g − Y M θ µµ ′ θ νν ′ ∇ µ ( K aνµ ′ ∂ ν ′ A a + J aµ ′ ∇ ν ∂ ν ′ A a )= e σ K a (cid:3) G A a + ∂ µ K a ∂ µ A a )= e σ K a (cid:3) G A a + R − J aν ∂ ν A a . (134)22herefore e − σ ∇ µ ∇ ν δ A G µν = − K a (cid:3) G A a + Ric[ G ] J aν ∂ ν A a . (135)Finally, we have 12 e − σ G µν δG µν = e − σ g µν J aµ ∂ ν A a = J aµ ∂ µ A a . (136)Therefore e − σ δ A Ric[ G ] = e − σ ( (cid:3) G + 12 Ric[ G ])( G µν δG µν ) − e − σ ∇ µ ∇ ν δG µν = (2 (cid:3) G + Ric[ G ])( J aµ ∂ µ A a ) + ( K a (cid:3) G A a − Ric[ G ] J aν ∂ ν A a )= K a (cid:3) G A a + 2 ∇ µ ( J aµ (cid:3) G A a ) , (137)where we used (cid:3) G ( J aµ ∂ µ A a ) = (cid:3) G J aµ ∂ µ A a + J aµ (cid:3) G ∂ µ A a + 2 K aµν ∇ µ ∇ ν A a = −
12 Ric[ G ] J aµ ∂ µ A a + J aµ ∂ µ (cid:3) G A a + Ric µν [ G ] J aµ ∂ ν A a + K a (cid:3) G A a = ∇ µ ( J aµ (cid:3) G A a ) − ( ∇ µ J aµ ) (cid:3) G A a + K a (cid:3) G A a = ∇ µ ( J aµ (cid:3) G A a ) (138)due to (132). Now we can use the equations of motion (107), which give ∇ µ ( J aµ (cid:3) G A a ) = 14 g Y M e σ/ ∇ µ g µν ∇ ρ ˜ T νρ = 14 g Y M e − σ/ ∇ µ ∇ ν ˜ T µν K a (cid:3) G A a = − g Y M e σ/ K a K a T = 12 g Y M e − σ/ R − T (139)recalling that J a K a = 0, as well as K a K a = − e − σ R − . (140)Putting these together, we finally arrive at δ A Ric[ G ] = 12 g Y M e σ/ (cid:16) R − T + ∇ µ ∇ ν ˜ T µν (cid:17) = 8 πG N (cid:0) T + R ∇ µ ∇ ν ˜ T µν (cid:1) πG N = e σ/ g Y M R = κe σ R . (141)23 eferences [1] J. Madore, “The Fuzzy sphere,” Class. Quant. Grav. (1992) 69.[2] H. Grosse, C. Klimcik and P. Presnajder, “Towards finite quantum field theory innoncommutative geometry,” Int. J. Theor. Phys. (1996) 231 [hep-th/9505175];H. Grosse, C. Klimcik and P. Presnajder, “Field theory on a supersymmetric lattice,”Commun. Math. Phys. , 155 (1997) [hep-th/9507074].[3] C. -S. Chu, J. Madore and H. Steinacker, “Scaling limits of the fuzzy sphere at oneloop,” JHEP (2001) 038 [hep-th/0106205].[4] D. Karabali, V. P. Nair and A. P. Polychronakos, “Spectrum of Schrodingerfield in a noncommutative magnetic monopole,” Nucl. Phys. B , 565 (2002)[hep-th/0111249].[5] S. Vaidya, “Scalar multi solitons on the fuzzy sphere,” JHEP (2002) 011[hep-th/0109102].[6] H. Steinacker, “Quantized gauge theory on the fuzzy sphere as random matrix model,”Nucl. Phys. B (2004) 66 [hep-th/0307075].[7] P. Castro-Villarreal, R. Delgadillo-Blando and B. Ydri, “A Gauge-invariant UV-IRmixing and the corresponding phase transition for U(1) fields on the fuzzy sphere,”Nucl. Phys. B , 111 (2005) [hep-th/0405201].[8] A. P. Balachandran, S. Kurkcuoglu and S. Vaidya, “Lectures on fuzzy and fuzzy SUSYphysics,” Singapore, Singapore: World Scientific (2007) 191 p. [hep-th/0511114].[9] R. Delgadillo-Blando, D. O’Connor and B. Ydri, “Matrix Models, Gauge Theory andEmergent Geometry,” JHEP (2009) 049 [arXiv:0806.0558 [hep-th]].[10] P. -M. Ho and M. Li, “Large N expansion from fuzzy AdS(2),” Nucl. Phys. B (2000) 198 [hep-th/0005268].[11] P. -M. Ho and M. Li, “Fuzzy spheres in AdS / CFT correspondence and holographyfrom noncommutativity,” Nucl. Phys. B (2001) 259 [hep-th/0004072].[12] E. Joung, J. Mourad and R. Parentani, “Group theoretical approach to quantum fieldsin de Sitter space. I. The Principle series,” JHEP (2006) 082 [hep-th/0606119];E. Joung, J. Mourad and R. Parentani, “Group theoretical approach to quantum fieldsin de Sitter space. II. The complementary and discrete series,” JHEP (2007) 030[arXiv:0707.2907 [hep-th]].[13] H. Steinacker, “Emergent Gravity and Noncommutative Branes from Yang-Mills Ma-trix Models,” Nucl. Phys. B (2009) 1 [arXiv:0806.2032 [hep-th]]; H. Steinacker,“Emergent Geometry and Gravity from Matrix Models: an Introduction,” Class.Quant. Grav. , 133001 (2010) [arXiv:1003.4134 [hep-th]].2414] H. Steinacker, “Gravity and compactified branes in matrix models,” JHEP (2012) 156 [arXiv:1202.6306 [hep-th]]; H. Steinacker, “The curvature of branes, cur-rents and gravity in matrix models,” JHEP (2013) 112 [arXiv:1210.8364 [hep-th]];[15] H. Steinacker, “On the Newtonian limit of emergent NC gravity and long-distancecorrections,” JHEP , 024 (2009) [arXiv:0909.4621 [hep-th]][16] J. D. Brown, M. Henneaux and C. Teitelboim, “Black Holes In Two Space-time Di-mensions,” Phys. Rev. D (1986) 319; R. B. Mann, A. Shiekh and L. Tarasov,“Classical And Quantum Properties Of Two-dimensional Black Holes,” Nucl. Phys. B (1990) 134.[17] V. Bargmann, “Irreducible unitary representations of the Lorentz group,” Ann. Math. (1947) 568.[18] Joe Repka, “Tensor products of unitary representations of SL ( R ),” Bull. Amer. Math.Soc. (1976) 930-932;A. van Tonder, “Cohomology and decomposition of tensor product representations ofSL(2,R),” Nucl. Phys. B (2004) 614 [hep-th/0212149],V. F. Molˇcanov, “Tensor products of unitary representations of the three-dimensionalLorentz group,” Math. USSR Izv. (1980) 113;R. P. Martin,“Tensor products for SL(2, k)”, Trans. Amer. Math. Soc. (1978),197[19] Wayne J. Holman and Lawrence C. Biedenharn, ”Complex angular momenta and thegroups SU (1 ,
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