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Signature Change in Matrix Model Solutions
Allen Stern - University of Alabama Chuang Xu - University of Alabama
Deposited 04/18/2019
Citation of published version:
Stern, A., Xu, C. (2018): Signature Change in Matrix Model Solutions. Physical Review D, 98(8). DOI: https://doi.org/10.1103/PhysRevD.98.086015
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP. Published by the American Physical Society. PHYSICAL REVIEW D 98, 086015 (2018)
Signature change in matrix model solutions
† A. Stern* and Chuang Xu (许闯) Department of Physics, University of Alabama, Tuscaloosa, Alabama 35487, USA
(!) (Received 9 September 2018; published 12 October 2018)
Various classical solutions to lower dimensional Ishibashi-Kawai-Kitazawa-Tsuchiya–like Lorentzian matrix models are examined in their commutative limit. Poisson manifolds emerge in this limit, and their associated induced and effective metrics are computed. Signature change is found to be a common feature of these manifolds when quadratic and cubic terms are included in the bosonic action. In fact, a single manifold may exhibit multiple signature changes. Regions with a Lorentzian signature may serve as toy models for cosmological spacetimes, complete with cosmological singularities, occurring at the signature change. The singularities are resolved away from the commutative limit. Toy models of open and closed cosmological spacetimes are given in two and four dimensions. The four-dimensional cosmologies are constructed from noncommutative complex projective spaces, and they are found to display a rapid expansion near the initial singularity.
DOI: 10.1103/PhysRevD.98.086015
I. INTRODUCTION constructed from these metrics. The singularities are resolved away from the commutative limit, where the description of Signature change is believed to be a feature of quantum the solution is in terms of representations of some matrix gravity [1–10]. It has been discussed in the context of string algebras. theory [6], loop quantum gravity [7,10], and causal dynami- As well as being of intrinsic interest, signature changing cal triangulation [8]. Recently, signature change has also matrix model solutions could prove useful for cosmology. been shown to result from certain solutions to matrix It has been shown that toy cosmological models can be equations [11–13]. These are the classical equations of constructed for regions of the manifolds where the metric motion that follow from Ishibashi, Kawai, Kitazawa, and has a Lorentzian signature. These regions can represent Tsuchiya (IKKT)-type models [14] with a Lorentzian back- both open and closed cosmologies, complete with cos- ground target metric. The signature change occurs in the mological singularities that occur at the signature changes. induced metrics of the continuous manifolds that emerge As stated above such singularities are resolved away from upon taking the commutative (or equivalently, continuum or the commutative limit. Furthermore, in [13], a rapid semiclassical) limit of the matrix model solutions. Actually, expansion, although not exponential, was found to occur as argued by Steinacker, the relevant metric for these immediately after the big bang singularity. emergent manifolds is not, in general, the induced metric, The previous examples of matrix models where signature but rather it is the metric that appears upon the coupling to change was observed include the fuzzy sphere embedded in matter [15]. The latter is the so-called effective metric of the a three-dimensional Lorentzian background [11], fuzzy emergent manifold, and it is determined from the symplectic 2 CP in an eight-dimensional Lorentzian background [12], structure that appears in the commutative limit, as well as the and noncommutative H4 in ten-dimensional Lorentz space- induced metric. Signature changes also occur for the effective time [13]. For the purpose of examining signature changes, metric of these manifolds, and, in fact, they precisely it is sufficient to restrict to the bosonic sector of the matrix coincide with the signature changes in the induced metric. models. In this article we present multiple additional The signature changes in the induced or effective metric examples of solutions to bosonic matrix models that exhibit correspond to singularities in the curvature tensor signature change. We argue that signature change is actually a common feature of solutions to IKKT-type *[email protected] matrix models with indefinite background metrics, in † [email protected] particular, when mass terms are included in the matrix model action. (Mass terms have been shown to result from Published by the American Physical Society under the terms of an IR regularization [16].) In fact, a single solution can the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to exhibit multiple signature changes. As an aside, it is known the author(s) and the published article’s title, journal citation, that there are zero mean curvature surfaces in three- and DOI. Funded by SCOAP3. dimensional Minkowski space that change from being
2470-0010=2018=98(8)=086015(20) 086015-1 Published by the American Physical Society A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018) spacelike to being timelike [17–21].1 The signature chang- change. The commutative limit of the deformed noncom- ing surfaces that emerge from solutions to the Lorentzian mutative CP2 solution has a region with Lorentzian sig- matrix models studied here do not have zero mean nature that describes a closed spacetime cosmology, curvature. complete with initial/final singularities. As with the non- As a warm-up, we review two-dimensional solutions to commutative H4 solution found in [13], these solutions the three-dimensional Lorentz matrix model, consisting of display an extremely rapid expansion near the cosmological a quartic (Yang-Mills) and a cubic term, and without a singularities. Also as in [13], the spacetimes emerging from quadratic (mass) term. Known solutions are noncommu- the deformed noncommutative CP1;1 and CP0;2 solutions 2 tative ðAÞdS [22–26] and the noncommutative cylinder expand linearly at late times. It suggests that these are [27–29]. They lead to a fixed signature upon taking the universal properties of 4d signature changing solutions to commutative limit. New solutions appear when the quad- IKKT-type matrix models. ratic term is included in the action. These new solutions The outline for this article is the following: In Sec. II we exhibit signature change. One such solution, found previ- review the noncommutative ðAÞdS2 and cylinder solutions ously, is the Lorentzian fuzzy sphere [11]. Others can be to the (massless) three-dimensional Lorentz matrix model. constructed by deforming the noncommutative AdS2 sol- We include the mass term in the matrix model action in ution. After taking the commutative limit of these matrix Sec. III, and examine the resulting signature changing solutions, one finds regions of the emergent manifolds matrix model solutions. The noncommutative CP1;1 and where the metric has a Lorentzian signature. In the case CP0;2 solutions to an eight-dimensional (massless) matrix of the Lorentzian fuzzy sphere, the Lorentzian region model (in the semiclassical limit) are examined in Sec. IV. crudely describes a two-dimensional closed cosmology, The mass term is added to the action in Sec. V, and there we complete with an initial and a final singularity. In the case study the resulting deformed noncommutative CP1;1, 2 of the deformation of noncommutative (Euclidean) ðAÞdS , CP0;2, and CP2 solutions. In Appendix A we list some the Lorentzian region describes a two-dimensional open properties of suð2; 1Þ in the defining representation. In cosmology. Appendix B we review a derivation of the effective metric Natural extensions of these solutions to higher dimen- and compute it for the examples of CP1;1 and CP0;2. sions are the noncommutative complex projective spaces [30–36]. Since we wish to recover noncompact manifolds, II. THREE-DIMENSIONAL as well as compact manifolds in the commutative limit, we LORENTZIAN MATRIX MODEL should consider the indefinite versions of these noncom- mutative spaces [37] as well as those constructed from We begin by considering the bosonic sector of the three- compact groups. For four-dimensional solutions, there are dimensional Lorentzian matrix model with an action then three such candidates: noncommutative CP2, CP1;1, consisting of a quartic (Yang-Mills) term and a cubic term: and CP0;2. The latter two solve an eight-dimensional 1 1 i (massless) matrix model with an indefinite background − μ ν ϵ μ ν λ SðXÞ¼ 2 Tr 4 ½Xμ;Xν ½X ;X þ3 a μνλX ½X ;X : metric, specifically, the suð2; 1Þ Cartan-Killing metric. g These solutions give a fixed signature after taking the ð2:1Þ commutative limit. So as with the two-dimensional sol- utions, the massless matrix model yields no signature Here Xμ, μ ¼ 0, 1, 2, are infinite-dimensional Hermitian change. Once again, new solutions appear when a mass matrices and a and g are constants. Tr denotes a trace, term is included, and they exhibit signature change, possibly indices μ; ν; λ; …, are raised and lowered with the Lorentz multiple signature changes. These solutions include defor- metric ημν ¼ diagð−; þ; þÞ, and the totally antisymmetric 2 1;1 0;2 2 mations of noncommutative CP , CP , and CP . A symbol ϵμνλ is defined such that ϵ012 ¼ 1. Extremizing the deformed noncommutative CP0;2 solution can undergo two action with respect to variations in Xμ leads to the classical signature changes, while a deformed noncommutative CP2 equations of motion solution can have up to three signature changes. Upon taking 1 1 ν ν λ the commutative limit, the deformed noncommutative CP ; ½½Xμ;Xν ;X þiaϵμνλ½X ;X ¼0: ð2:2Þ and CP0;2 solutions have regions with Lorentzian signature that describe expanding open spacetime cosmologies, com- The equations of motion (2.2) are invariant under the plete with a big bang singularity occurring at the signature following: (i) unitary “gauge” transformations, Xμ → UXμU†, where U is an infinite dimensional unitary matrix, 1We thank J. Hoppe for bringing this to our attention. μ μ ν 2 (ii) 2 þ 1 Lorentz transformations X → L νX , where As state above, we assume the background metric to be the 3 3 suð2; 1Þ Cartan-Killing metric. Noncommutative CP2 and its L is a × Lorentz matrix, and deformations were shown to solve an eight-dimensional Lor- (iii) translations in the three-dimensional Minkowski entzian matrix model with a different background metric in [12]. space Xμ → Xμ þ vμ1, where 1 is the unit matrix.
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Well known solutions to these equations are non- where −∞ < τ < ∞, 0 ≤ σ < 2π. Using this para- commutative ðAÞdS2 [22–26] and the noncommutative metrization we obtain the following Lorentzian in- cylinder [27–29]. Both are associated with unitary irre- duced metric on the surface3: ducible representations of three-dimensional Lie algebras. 2 2 2 − τ2 2 τ σ2 Noncommutative ðAÞdS corresponds to unitary irreduc- ds ¼ r ð d þ cosh d Þ: ð2:7Þ ible representations of suð1; 1Þ, while the noncommutative The Poisson brackets (2.5) are recovered upon writing cylinder corresponds to unitary irreducible representations of the two-dimensional Euclidean algebra E2. Thus the 1 fτ; σg¼ : ð2:8Þ former solution is defined by r cosh τ λ μ 2 ½Xμ;Xν ¼iaϵμνλX ;XμX fixed; ð2:3Þ (2) Noncommutative Euclidean ðAÞdS . A negative μ − 2 while the latter is Casimir yields the constraint xμx ¼ r in the commutative limit. This defines a two-sheeted hy- 2 0 perboloid corresponding to the Euclidean version of ½X0;X ¼ aX ; ½Xþ;X− ¼ ;XþX− fixed; ð2:4Þ de Sitter (or anti–de Sitter) space, Euclidean ðAÞdS2 2 2;0 where X ¼ X1 iX2. Noncommutative ðAÞdS preserves (or H ). A parametrization of the upper hyper- the Lorentz symmetry (ii) of the equations of motion, while boloid (x0 > 0)is the noncommutative cylinder breaks the symmetry to the 0 1 0 1 two-dimensional rotation group. Noncommutative AdS2 was 0 τ B x C B cosh C recently shown to be asymptotically commutative, and the B 1 C B C @ x A ¼ r@ sinh τ cos σ A; ð2:9Þ holographic principlewas applied to map a scalar field theory 2 τ σ on noncommutative AdS2 to a conformal theory on the x sinh sin boundary [26]. −∞ τ ∞ 0 ≤ σ 2π The commutative (or equivalently, continuum or semi- where again < < , < . Now the classical) limit for these two solutions is clearly a → 0.Thusa induced metric on the surface has a Euclidean plays the role of ℏ of quantum mechanics, and for conven- signature ℏ ience we shall make the identification a ¼ and then take the ds2 ¼ r2ðdτ2 þ sinh2 τdσ2Þ: ð2:10Þ limit ℏ → 0. In the limit, functions of the matrices Xμ are replaced by functions of commutative coordinates xμ, Upon assigning the Poisson brackets and to lowest order in ℏ, commutators of functions of Xμ 1 are replaced by iℏ times Poisson brackets of the correspond- τ σ f ; g¼ τ ð2:11Þ ing functions of xμ, ½FðXÞ; GðXÞ → iℏfFðxÞ; GðxÞg. r sinh So in the commutative limit of the noncommutative we again recover the suð1; 1Þ Poisson bracket ðAÞdS2 solution, Eq. (2.3) defines a two-dimensional algebra (2.5). hyperboloid with an suð1; 1Þ Poisson algebra The commutative limit of the noncommutative cylinder solution (2.4) is obviously the cylinder. The Casimir ϵ λ fxμ;xνg¼ μνλx : ð2:5Þ for the two-dimensional Euclidean algebra goes to 1 2 2 2 2 Two different geometries result from the choice of sign of ðx Þ þðx Þ ¼ r , while the limiting Poisson brackets 0 1 −2 2 0 2 2 1 1 2 0 the Casimir in (2.3). The positive sign is associated are fx ;x g¼ x , fx ;x g¼ x , fx ;x g¼ . A para- with noncommutative ðAÞdS2, while the negative sign is metrization in terms of polar coordinates 2 0 1 0 1 associated with noncommutative Euclidean ðAÞdS .We 0 τ describe them as follows: B x C B C 2 B 1 C B C (1) Noncommutative ðAÞdS . A positive Casimir yields @ x A ¼ @ r cos σ A ð2:12Þ μ 2 the constraint xμx ¼ r in the commutative limit, x2 r sin σ which defines two-dimensional de Sitter (or anti–de 2 1 1 Sitter) space, ðAÞdS (or H ; ). r in this semiclassical yields the Lorentzian induced metric solution, and the ones that follow, denotes a constant length scale, r>0. A global parametrization for ds2 ¼ −dτ2 þ r2dσ2; ð2:13Þ ðAÞdS2 is given by 0 1 0 1 and the Poisson algebra is recovered for fτ; σg¼2. 0 τ B x C B sinh C B 1 C B C @ x A ¼ r@ cosh τ cos σ A; ð2:6Þ 3AdS2 and dS2 are distinguished by the definition of the x2 cosh τ sin σ timelike direction on the manifold. For the former, the time parameter corresponds to σ, and for the latter, it is τ.
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The above solutions admit either Euclidean or ðx0Þ2 þðx1Þ2 þðx2Þ2 ¼ r2; fx0;x1g¼x2; Lorentzian induced metrics after taking the commutative fx1;x2g¼x0; fx2;x0g¼x1: ð3:6Þ limit. The signature for any of these particular solutions is fixed. Below we show that the inclusion of a mass These Poisson brackets solve the Lorentzian equations (3.4) term in the action allows for solutions with signature provided that α − 1 and β −1. The solution obviously change. ¼ 2 ¼ does not preserve the Lorentz symmetry (ii) of the equations of motion. One can introduce a spherical coor- III. INCLUSION OF A MASS TERM dinate parametrization d IN THE 3 MATRIX MODEL 0 1 0 1 0 θ We next add a quadratic contribution to the three- B x C B cos C B 1 C B C dimensional Lorentzian matrix model action (2.1): @ x A ¼ r@ sin θ cos ϕ A; ð3:7Þ x2 sin θ sin ϕ 1 1 μ ν i μ ν λ SðXÞ¼ Tr − ½Xμ;Xν ½X ;X þ aϵμνλX ½X ;X g2 4 3 0 ≤ ϕ < 2π, 0 < θ < π. Then the Poisson brackets in (3.6) 1 b μ are recovered for fθ; ϕg¼ csc θ. The induced invariant þ X Xμ : ð3:1Þ r 2 length which one computes from the Lorentzian back- 2 μ ground, ds ¼ dx dxμ, does not give the usual metric for a As stated in the Introduction, quadratic terms have been sphere. Instead, one finds shown to result from an IR regularization [16]. The μ equations of motion resulting from variations of X are now ds2 ¼ r2ðcos 2θdθ2 þ sin2θdϕ2Þ: ð3:8Þ
ν ν λ ½½Xμ;Xν ;X þiaϵμνλ½X ;X þbXμ ¼ 0: ð3:2Þ In addition to the coordinate singularities at the poles, there π 3π are singularities in the metric at the latitudes θ ¼ 4 and 4 . As in this article we shall only be concerned with solutions The Ricci scalar is divergent at these latitudes. The metric 0 θ π in the commutative limit, ℏ → 0, we may as well take the tensor has a Euclidean signature for < < 4 and 3π π 3π limit of these equations. In order for the cubic and quadratic 4 < θ < π, and a Lorentzian signature for 4 < θ < 4 . terms to contribute in the commutative limit we need that a The regions are illustrated in Fig. 1. The Lorentzian regions and b vanish in the limit according to of the fuzzy sphere solutions have both an initial and a final
a → ℏα;b→ ℏ2β; ð3:3Þ where α and β are nonvanishing and finite. Equation (3.2) reduces to
ν ν λ −ffxμ;xνg;x g − αϵμνλfx ;x gþβxμ ¼ 0: ð3:4Þ
The AdS2 and Euclidean AdS2 solutions, which are associated with the suð1; 1Þ Poisson algebra (2.5), survive when the mass term is included provided that the constants α and β are constrained by
β ¼ 2ð1 − αÞ: ð3:5Þ
In the limit where the mass term vanishes, β ¼ 0 and α ¼ 1, we recover the solutions of the previous section. On the other hand, the noncommutative cylinder only solves the β → 0 equations in the limit of zero mass . FIG. 1. Commutative limit of the Lorentzian fuzzy sphere. π 3π The mass term allows for new solutions, which have no Singularities in the metric appear at θ ¼ 4 and 4 , and the signature β → 0 limit. One such solution is the fuzzy sphere of the metric changes at these latitudes. These latitudes are embedded in the three-dimensional Lorentzian back- associated with singularities in the Ricci scalar. The metric tensor π 3π ground, which was examined in [11]. In the commutative has a Euclidean signature for 0 < θ < 4 and 4 < θ < π (darker π 3π limit it is defined by regions), and a Lorentzian signature for 4 < θ < 4 (lighter region).
086015-4 SIGNATURE CHANGE IN MATRIX MODEL SOLUTIONS PHYS. REV. D 98, 086015 (2018) singularity and thus crudely describe a two-dimensional closed cosmology. The singularities are resolved away from the commutative limit, where the fuzzy sphere is expressed in terms of N × N Hermitian matrices. Axially symmetric deformations of the fuzzy sphere are also solutions to the Lorentzian matrix model [11].
Other sets of solutions to the Lorentzian matrix model 2 which have no β → 0 limit are deformations of the non- commutative AdS2 and the Euclidean AdS2 solutions. Like the fuzzy sphere solution, they break the Lorentz symmetry (ii) of the equations of motion, but preserve spatial rota- 0 tional invariance. Again, we shall only be concerned with the commutative limit of these solutions: (1) Deformed noncommutative AdS2. Here we replace (2.6) by -2 0 1 0 1 0 τ B x C B sinh C B 1 C B C @ x A ¼ r@ ρ cosh τ cos σ A: ð3:9Þ -4 -2 0 2 4 x2 ρ cosh τ sin σ FIG. 2. Deformed AdS2 solution with r ¼ 1, ρ ¼ 1.15. The ρ 0 −1 1 > is the deformation parameter. We again spacetime singularities occur at τ ¼ τ ¼ tanh j ρ j. The τ σ assume the Poisson bracket (2.8) between and . lighter region has a Lorentzian signature, and the darker region 2 Substituting (3.9) into (3.4) gives β ¼ 2ρ ð1 − αÞ¼ has a Euclidean signature. 1 þ ρ2 − 2α. It is solved by the previous undeformed AdS2 solution, ρ2 ¼ 1 with (3.5), along with new provided that the relations (3.10) again hold. The solutions that allow for arbitrary ρ > 0, provided that induced invariant interval on the surface is now 1 α β ρ2 ds2 ¼ r2sinh2τððρ2coth2τ − 1Þdτ2 þ ρ2dσ2Þ: ð3:13Þ ¼ 2 ; ¼ : ð3:10Þ ρ2 1 τ τ For < there is a singularity at ¼ þ ¼ Using the parametrization (3.9), the induced invari- −1 ant interval on the surface is now tanh jρj that is associated with a signature change. τ τ For < þ the signature of the induced metric is 2 2 2τ −1 ρ2 2τ τ2 ρ2 σ2 τ τ ds ¼ r cosh ðð þ tanh Þd þ d Þ: Euclidean, while for > þ the signature of the induced metric is Lorentzian. Figure 3 gives a plot of ð3:11Þ deformed hyperboloid in the three-dimensional em- 2 1 ρ 0 85 ρ > 1 bedding space for r ¼ , ¼ . . The deformed For the induced metric tensor possesses 2 τ τ −1 1 Euclidean AdS solution has only an initial (big spacetime singularities at ¼ ¼ tanh j ρ j, bang) singularity that appears in the commutative which are associated with two signature changes. τ τ τ τ For > þ and < − the signature of the induced τ τ τ metric is Euclidean, while for − < < þ the signature of the induced metric is Lorentzian. Figure 2 is a plot of deformed AdS2 in the three-dimensional embedding space for r ¼ 1, ρ ¼ 1.15. (2) Deformed noncommutative Euclidean AdS2.We now deform the upper hyperboloid given in (2.9) to 0 1 0 1 2 0 τ B x C B cosh C B 1 C B C @ x A ¼ r@ ρ sinh τ cos σ A; ð3:12Þ -2 0 2 x2 ρ sinh τ sin σ FIG. 3. Deformed Euclidean AdS2 solution with r ¼ 1, τ ρ 0 85 τ τ −1 ρ while retaining the Poisson bracket (2.11) between ¼ . . A spacetime singularity occurs at ¼ þ ¼tanh j j. and σ:ρ again denotes the deformation parameter. The lighter region has a Lorentzian signature, and the darker region Equation (3.12) with ρ ≠ 0 is a solution to (3.4) has a Euclidean signature.
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i 1 limit, and so, crudely speaking, the Lorentzian z zi ¼ ; ð4:4Þ region describes an open two-dimensional cosmol- ogy. The singularity is resolved away from the while for CP0;2 the constraint can be written as commutative limit. i −1 z zi ¼ . ð4:5Þ CP1;1 CP0;2 IV. AND SOLUTIONS For both cases, the Poisson brackets (4.3) become Concerning the generalization to four dimensions, a i − δi i j 0 natural approach would be to examine noncommutative fz ;zj g¼ i j; fz ;z g¼fzi ;zj g¼ : ð4:6Þ CP2 [30–36]. Actually, if we wish to recover noncompact 1 1 0 2 manifolds in the commutative limit, we should consider the CP ; and CP ; can also be described in terms of orbits indefinite versions of noncommutative CP2; noncommu- on SUð2; 1Þ. Below we review some properties of the Lie tative CP1;1 and CP0;2. In this section we show that algebra suð2; 1Þ. One can write down the defining repre- 1 1 0 2 2 1 3 3 noncommutative CP ; and CP ; are solutions to an sentation for suð ; Þ in terms of traceless × matrices, λ˜ 1 2 … 8 eight-dimensional matrix model with an indefinite back- a, a ¼ ; ; ; , which are analogous to the Gell-Mann λ 3 ground metric. As with our earlier result, we find no matrices a spanning suð Þ. We denote matrix elements by λ˜ i … 1 3 signature change in the absence of a mass term in the ½ a j;i;j; ¼ , 2, 3. Unlike suð Þ Gell-Mann matrices, ˜ action. A mass term will be included in the following λa are not all Hermitian, but instead, satisfy section. Here, we begin with some general properties of 1 1 0 2 noncommutative CP ; and CP ; in the semiclassical limit, ˜ C C ˜† λaη ¼ η λa: ð4:7Þ and then construct an eight-dimensional matrix model for which they are solutions. They are given in terms of the standard Gell-Mann matrices ˜ in Appendix A. The commutation relations for λa are A. Properties ½λ˜ ; λ˜ ¼2if˜ λ˜c; ð4:8Þ Noncommutative CPp;q was studied in [37]. Here we a b abc p;q shall only be interested in its semiclassical limit. CP are where indices a; b; c; …, are raised and lowered using the 2q;2pþ1 1 hyperboloids H mod S . They can be defined in Cartan-Killing metric on the eight-dimensional space terms of p þ q þ 1 complex embedding coordinates zi, 1 … 1 2q;2pþ1 i ¼ ; ;pþ q þ , satisfying the H constraint η ¼ diagð1; 1; 1; −1; −1; −1; −1; 1Þ: ð4:9Þ Xpþ1 pXþqþ1 − 1 ˜ 2 1 zi zi zi zi ¼ ; ð4:1Þ fabc for suð ; Þ are totally antisymmetric. Their values, i¼1 i¼pþ2 along with some properties of suð2; 1Þ, are given in Appendix A. along with the identification CP0;2 is the coset space SUð2; 1Þ=Uð2Þ. Using the ∼ iβ conventions of Appendix A, it is spanned by adjoint zi e zi: ð4:2Þ ˜ orbits in suð2; 1Þ through λ8 and consists of elements p;q ˜ −1 ˜ CP can equivalently be defined as the coset space gλ8g ;g∈ SUð2; 1Þ. The little group of λ8 is Uð2Þ, which 1 ˜ ˜ ˜ ˜ 1;1 SUðp þ ;qÞ=Uðp; qÞ. For the semiclassical limit of non- is generated by λ1; λ2; λ3; λ8. On the other hand, CP is the p;q commutative CP we must also introduce a compatible coset space SUð2; 1Þ=Uð1; 1Þ. It corresponds to orbits Poisson structure. For this we take through 0 1 −iδij; if i; j ¼ 1; …;pþ 1; −2 ffiffiffi fz ;z g¼ ; p i j δ 2 … 1 ˜ 1 B C 3 ˜ 1 ˜ i ij; if i; j ¼ p þ ; ;pþ q þ Λ8 ¼ pffiffiffi @ 1 A ¼ − λ3 − λ8; ð4:10Þ 3 2 2 ð4:3Þ 1