Signature Change in Matrix Model Solutions

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 noncommutative 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;Xν½X ;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 solutions, 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.

086015-2 SIGNATURE CHANGE IN MATRIX MODEL SOLUTIONS PHYS. REV. D 98, 086015 (2018)

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;Xν¼iaϵμνλX ;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;X¼ 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.

086015-3 A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018)

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;Xν½X ;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 which have no β → 0 limit are deformations of the noncommutative 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 rotational invariance. Again, we shall only be concerned with the commutative limit of these solutions: (1) Deformed noncommutative AdS2. Here we replace (2.6) by 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 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 ¼tanh 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 ¼tanh 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 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 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.

086015-5 A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018)

i 1 limit, and so, crudely speaking, the Lorentzian z zi ¼ ; ð4:4Þ region describes an open two-dimensional cosmology. 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 ;zg¼ ; 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

1;1 ˜ −1 ˜ while all other Poisson brackets amongst zi and zi vanish. CP ¼fgΛ8g ;g∈ SUð2; 1Þg. The little group of Λ8 is Then one can regard (4.1) as the first class constraint that Uð1; 1Þ, which is generated by generates the phase equivalence (4.2). 1;1 0;2 pffiffiffi In specializing to CP and CP , it is convenient 1 3 ηC 1 1 −1 Λ˜ λ˜ λ˜ Λ˜ 0 1 −1 − λ˜ λ˜ to introduce the metric ¼ diagð ; ; Þ on the 8; 6; 7; 3 ¼ diagð ; ; Þ¼ 2 3 þ 2 8: three-dimensional complex space spanned by zi, i ¼ 1, i ηC ij a i 2, 3. Then writing z ¼ð Þ zj, the constraint (4.1) for Next, we can construct eight real coordinates x from z 1;1 CP becomes and zi using

086015-6 SIGNATURE CHANGE IN MATRIX MODEL SOLUTIONS PHYS. REV. D 98, 086015 (2018)

˜a xa ¼ z½λ i zj: ð4:11Þ 2 a 2 i j ds ¼ dx dxa ¼ 4ððzz¯ Þðdzdz¯ Þ − jzdz¯ j Þ; ð4:17Þ They are invariant under the phase transformation (4.2) and where zdz¯ ¼ zdzi, dzdz¯ ¼ dzdzi, and we have used span a four-dimensional manifold. Using (A6), the con- i i dðzz¯ Þ¼0. Equation (4.17) is the Fubini-Study metric straints on the coordinates are written on a noncompact space. 4 2 We next examine the induced metric tensor on a local xax ¼ ; d˜ xbxc ¼ x ; ð4:12Þ a 3 abc 3 a coordinate patch. We choose the local coordinates ðζ1; ζ2Þ, defined by where one takes the upper sign in the second equation 1;1 0;2 ˜ 1 2 for CP and the lower sign for CP . dabc is totally z z 3 ζ1 ¼ ; ζ2 ¼ ;z≠ 0; ð4:18Þ symmetric; the values are given in Appendix A. From (4.6), z3 z3 xa satisfy an suð2; 1Þ Poisson bracket algebra along with their complex conjugates. These coordinates ˜ c fxa;xbg¼2fabcx : ð4:13Þ respect the equivalence relation (4.2). In the language of constrained Hamilton formalism, they are first class var- 2 2 iables. From their definition it follows that jζ1j þjζ2j − 3 −2 3 2 3 B. Eight-dimensional matrix model 1 ¼jz j and zdz¯ ¼jz j ðζ1dζ1 þ ζ2dζ2Þd log z , 1;1 0;2 It is now easy to construct an eight-dimensional IKKT- where the upper [lower] sign applies for CP [CP ]. type matrix model for which (4.13) is a solution, at least in Substituting into (4.17) gives the induced metric tensor on the commutative limit. As before we only consider the the coordinate patch bosonic sector, spanned by eight infinite-dimensional 1 1 Hermitian matrices X , with indices raised and lowered 2 a ζ ζ ds ¼ gζuζv d ud v with the indefinite flat metric ηab. In analogy with the three- 4 2 dimensional model in (2.1), take the action to consist of a ζ 2 ζ 2 ζ ζ ζ ζ 2 jd 1j þjd 2j − j 1d 1 þ 2d 2j quartic term and a cubic term: ¼ 2 2 2 2 2 : ð4:19Þ jζ1j þjζ2j − 1 ðjζ1j þjζ2j − 1Þ 1 1 i − a b ˜ a b c 1;1 0;2 SðXÞ¼ 2 Tr 4 ½Xa;Xb½X ;X þ3 afabcX ½X ;X : It has the same form for both CP and CP . Because g 2 2 0;2 jζ1j þjζ2j − 1 < 0 for the latter, CP has a Euclidean ð4:14Þ signature. The Poisson brackets (4.6) can be projected down to the local coordinate patch as well. The result is The cubic term appears ad hoc, and we remark that it is 2 2 actually unnecessary for the purpose of finding solutions fζu; ζvg¼iðjζ1j þjζ2j − 1Þðζuζv − δuvÞ; when a quadratic term is introduced instead. We consider ζ ζ ζ ζ 0 1 2 quadratic terms in Sec. V. On the other hand, the cubic term f u; vg¼f u; vg¼ ;u;v¼ ; : ð4:20Þ leads to a richer structure for the space of solutions, and it is 1;1 for that reason we shall consider it. Once again, the upper [lower] sign applies for CP 0;2 The equations of motion following from (4.14) are [CP ]. The resulting symplectic two-form is Kähler:

b ˜ b c ½½Xa;Xb;X þiafabc½X ;X ¼0: ð4:15Þ i Ω ¼∓ gζ ζ dζ ∧ dζ : ð4:21Þ 2 u v u v They are invariant under unitary “gauge” transformations, SUð2; 1Þ transformations, and translations. Assuming that Next we rewrite the induced metric and symplectic two- the constant a behaves as in (3.3) in the commutative limit form using three Euler-like angles ðθ; ϕ; ψÞ, 0 ≤ θ < π, leads to 0 ≤ ϕ < 2π, 0 ≤ ψ < 4π, along with one real variable τ, −∞ < τ < ∞. We treat CP1;1 and CP0;2 separately: b ˜ b c 1;1 −ffxa;xbg;x g − αfabcfx ;x g¼0: ð4:16Þ (1) CP . Now write α 2 θ The Poisson brackets (4.13) solve these equations for ¼ . iðψþϕÞ=2 ζ1 ¼ e coth τ cos ; They describe a CP1;1 or a CP0;2 solution, the choice 2 depending on the sign in the second constraint in (4.12). θ ζ iðψ−ϕÞ=2 τ For either solution, we can project the eight-dimensional 2 ¼ e coth sin 2 ; ð4:22Þ η ¯ i 1 flat metric down to the surface zz ¼ zi z ¼ , in order to obtain the induced metric. Once again, i ¼ 1; 2; 3. Using which is consistent with the requirement that 2 2 the Fierz identity (A6), we get jζ1j þjζ2j − 1 > 0. The induced metric in these

086015-7 A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018)

3 4 coordinates has the Taub-(Newman, Unti, Tambur- Einstein equations with cosmological constant Λ ¼ 2. 2 ino) NUT form (which was also true for the CP Obviously, the metric tensors do not exhibit signature solution [12]) change. In both cases, the sign of the determinant of the 2 metric tensor, det g ¼ gττgψψðgθθ sin θÞ , is positive (away 2 τ2 θ2 2 θ ϕ2 ds ¼ gττd þ gθθðd þ sin d Þ from coordinate singularities). 2 þ gψψðdψ þ cos θdϕÞ : ð4:23Þ The above discussion utilized the induced metric tensor. However, the relevant metric in the semiclassical limit for a We get matrix model solution is not, in general, the induced metric, but rather it is the metric that appears in the coupling to 2 2 2 gττ ¼ −4;gθθ ¼ cosh τ;gψψ ¼ −cosh τsinh τ; matter [15]. This is the so-called “effective” metric tensor, which we here denote by γμν. It can be determined from the ð4:24Þ μν induced metric gμν and the symplectric matrix Θ using with the other nonvanishing components of the 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 induced metric being gϕϕ gψψ cos θ gθθ sin θ μν μν ¼ þ j det γjγ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ΘTgΘ : ð4:29Þ and gψϕ ¼ gψψ cos θ. The result indicates that there j det Θj are two spacelike directions and two timelike directions. The symplectic two-form in terms of these It follows that j det γj¼jdet gj, and we can use this coordinates is identification to determine the effective metric from the induced metric. We review a derivation of (4.29) in Ω 1 1 − τ τ τ ∧ ψ θ ϕ CP ; ¼ sinh cosh d ðd þ cos d Þ Appendix B. In two dimensions, it is known that the 1 effective metric is identical to the induced metric, γμν ¼ gμν 2τ θ θ ∧ ϕ þ 2 cosh sin d d [38]. This is also the case for the CP1;1 and CP0;2 solutions, 1 as is shown in Appendix B, and so all the previous results − 2τ ψ θ ϕ ¼ 2 dðcosh ðd þ cos d ÞÞ: ð4:25Þ that followed from the induced metric also apply for the effective metric. On the other hand, for the solutions of the (2) CP0;2. Here choose next section, in addition to finding signature change, we find that the effective metric and the induced metric for any θ ζ iðψþϕÞ=2 τ particular emergent manifold are in general distinct. 1 ¼ e tanh cos 2 ; θ ζ iðψ−ϕÞ=2 τ 2 ¼ e tanh sin 2 ; ð4:26Þ V. INCLUSION OF A MASS TERM IN THE 8d MATRIX MODEL which is consistent with the inequality ζ 2 ζ 2 − 1 0 In analogy to Sec. III, we now add a mass term to the j 1j þj 2j < . The resulting induced metric matrix model action (4.14), again has the Taub-NUT form (4.23). In comparing with (4.24), results differ for the gθθ component, 1 1 − a b SðXÞ¼ 2 Tr ½Xa;Xb½X ;X 2 2 2 g 4 gττ ¼ −4;gθθ ¼ −sinh τ;gψψ ¼ −cosh τsinh τ; i 4:27 ˜ a b c 6˜ a ð Þ þ 3 afabcX ½X ;X þ bXaX : ð5:1Þ 2 2 where again gϕϕ ¼ gψψ cos θ þ gθθ sin θ and The matrix equations of motion become gψϕ ¼ gψψ cos θ. The induced metric now has a

Euclidean signature, and the symplectic two-form is b ˜ b c ˜ ½½Xa;Xb;X þiafabc½X ;X þ12bXa ¼ 0: ð5:2Þ

Ω 0 2 − τ τ τ ∧ ψ θ ϕ CP ; ¼ sinh cosh d ðd þ cos d Þ In the semiclassical limit ℏ → 0, we take a → ℏα, along 1 ˜ → ℏ2β˜ 2τ θ θ ∧ ϕ with b . Then (5.2) goes to þ 2 sinh sin d d 1 − b − α˜ b c 12β˜ 0 − 2τ ψ θ ϕ ffxa;xbg;x g fabcfx ;x gþ xa ¼ : ð5:3Þ ¼ 2 dðsinh ðd þ cos d ÞÞ: ð4:28Þ

Both metric tensors (4.24) and (4.27) (including the 1;1 2 corresponding results for gϕϕ and gψϕ) describing CP 4CP is also a solution to the sourceless Einstein equations 0;2 3 and CP , respectively, are solutions to the sourceless with cosmological constant Λ ¼ 2 [12].

086015-8 SIGNATURE CHANGE IN MATRIX MODEL SOLUTIONS PHYS. REV. D 98, 086015 (2018)

These equations are solved by (4.13) for Upon requiring γ½β˜ to be real, we obtain three disconnected intervals (i)–(iii) in β˜: α 2 1 β˜ : 5:4 ¼ ð þ Þ ð Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffi pffiffiffi 1;1 0;2 β˜ ≤ 6 − 3 6 − 98 40 6 ≈−0 746 Thus CP and CP are solutions to the massive matrix ðiÞ 2 þ . ; β˜ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi model. In the limit where the mass term vanishes, ¼ 1 pffiffiffi pffiffiffi α 2 6 − 3 6 98 40 6 ≈−0 603 ≤ β˜ and ¼ , we recover the solutions of the previous section. ðiiÞ 2 þ þ . CP1;1 and CP0;2 solutions also persist in the absence of the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffi pffiffiffi cubic term in the matrix model action (5.1). For this we ≤ 6 3 6 − 98 40 6 ≈−0 325 2 þ þ . ; need α 0 and β˜ −1. The mass term allows for other qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ¼ ffiffiffi ffiffiffi solutions that have no β → 0 limit. Among these solutions 1 p p ðiiiÞ 6 þ 3 6 þ 98 þ 40 6 ≈ 13.67 ≤ β˜: ð5:9Þ are the deformations of CP1;1 and CP0;2, as well as 2 deformations of CP2, which we discuss in the following μ α ν subsections. We further restrict to be real. (Reality of and then follows.) For solution (5.7), this reduces the acceptable regions in β˜ to A. Deformations of CP1;1 and CP0;2 1;1 0;2 For deformations of CP and CP we modify the ði0Þ − 3.414 ≲ β˜ ≲ −0.746; ansatz (4.11) to ðii0Þ − 0.603 ≲ β˜ ≲ −0.586; μ λ˜ i j λ˜ i j ν λ˜ i j ˜ x1–3 ¼ zi ½ 1−3 jz ;x4–7 ¼zi ½ 4−7 jz ;x8 ¼ zi ½ 8 jz ; ðiii0Þ 13.67 ≲ β: ð5:10Þ ð5:5Þ When β˜ ¼ −0.6, we recover the undeformed case μ¼ν¼1 where μ and ν are deformation parameters, which we shall (along with α ¼ 0.8). Therefore, matrix solutions in the restrict to be real. This is a solution to Eq. (5.3) provided range ðii0Þ can be regarded as continuous deformations of that the following relations hold amongst the parameters: the undeformed solutions, while those in the ranges ði0Þ and (iii) cannot be continuously connected to the undeformed 1 solutions. 2μ − α μ2 3μβ˜ 0 ð Þ þ 2 þ ¼ ; In addition to the family of solutions given in (5.7) and (5.8), Eq. (5.6) have the simple solution: μ2 þ ν2 þ 2 − αðμ þ νÞþ4β˜ ¼ 0; 2ν − α 2νβ˜ 0 3 2 þ ¼ : ð5:6Þ α ν 0 β˜ − μ2 ¼ ¼ ; ¼ 5 ; ¼ 5 : ð5:11Þ These relations reduce to (5.4) when μ ¼ ν ¼ 1, and so we recover undeformed CP1;1 and CP0;2 in this limit. For It is a solution for the case where the cubic term in the generic values of the parameters, there are nontrivial matrix model action (5.1) is absent. From (5.5), ν ¼ 0 solutions to these algebraic relations, which can be implies that the projection of the solution along the eighth 0 expressed as functions of the mass parameter β˜. For a direction vanishes, x8 ¼ . This solution is not contained in particular choice of signs, (5.7) and (5.8). The ansatz (5.5) for μ and ν not both equal to one leads to β˜ 2 − β˜ − 1 − γ½β˜ two types of solutions: α ¼ 2μ ; (1) Deformed CP1;1, where the complex coordinates zi 2β˜ þ 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi satisfy the constraint (4.4), and 0;2 β˜ 3 − 4β˜ 2 − 6β˜ þ βγ˜ ½β˜ − 2 (2) Deformed CP , where the complex coordinates μ ¼ ; i 2 β˜ 2 4β˜ 2 z satisfy (4.5). ð þ þ Þ We next compute the induced metric for these two types of α ν ¼ ; ð5:7Þ solutions. 2ð1 þ β˜Þ 1. Induced metric where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The induced metric is again computed by projecting the γ β˜ β˜ 4 − 12β˜ 3 − 22β˜ 2 − 12β˜ − 2 eight-dimensional flat metric (4.9) onto the surface. From ½ ¼ : ð5:8Þ the ansatz (5.5) we get

086015-9 A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018)

2 a 2 2 2 2 2 2 2 ds ¼ dx dxa ¼ 4ðjz1j þjz2j − jz3j Þðjdz1j þjdz2j − jdz3j Þ − 4jz1dz1 þ z2dz2 − z3dz3j 2 2 2 2 2 2 2 þ 4ðμ − 1Þðjz1j þjz2j Þðjdz1j þjdz2j Þþðμ − 1Þðz1dz1 þ z2dz2 − z1dz1 − z2dz2Þ 1 ν2 − 1 2 2 2 þ 3 ð Þðz1dz1 þ z2dz2 þ z3dz3 þ z1dz1 þ z2dz2 þ z3dz3Þ : ð5:12Þ

We have not yet specialized to the two cases 1. and 2. The result (5.12) can be rewritten in terms of the local coordinates ðζ1; ζ2Þ, defined in (4.18), according to

2 2 2 2 2 2 ds ¼ 4jz3j ð−jz3j jΞj ðjdζ1j þjdζ2j ÞÞ 2 2 −2 2 2 2 2 þ 4ðμ − 1Þðjz3j 1Þðð1 jz3j Þjdz3j þjz3j ðjdζ1j þjdζ2j ÞþΞz3dz3 þ Ξ z3dz3Þ 2 2 −2 2 2 2 2 þðμ − 1Þðjz3j ðΞ − Ξ Þþð1 jz3j Þðz3dz3 − z3dz3ÞÞ þðν − 1Þðdjz3j Þ ; ð5:13Þ

2 2 where Ξ ¼ ζ1dζ1 þ ζ2dζ2 and we have used jζ1j þjζ2j ¼ The remaining nonvanishing components of the −2 1 jz3j . The upper [lower] sign applies for deformed induced metric are again obtained from gϕϕ ¼ 1;1 0;2 2 2 CP [CP ]. The expression (5.13) simplifies after making gψψ cos θ þ gθθ sin θ and gψϕ ¼ gψψ cos θ. The un- 1 1 the gauge choice that z3 is real, which we shall do below. deformed CP ; induced metric tensor in (4.24) is The signature of the induced metric becomes more recovered from (5.14) upon setting μ ¼ ν ¼ 1.This evident after expressing it in terms of the three Euler-like limit thus corresponds to there being two spacelike θ ϕ ψ τ angles ; ; , along with parameter spanning Rþ,aswe directions and two timelike directions, with did in Sec. IV for the undeformed metrics. For this we signðgττ;gθθÞ¼ð−; þÞ and det g>0 (away from now specialize to the two cases: 1. deformed CP1;1 and 2. coordinate singularities). The same two spacelike deformed CP0;2. directions and two timelike directions appear in the (1) Deformed CP1;1. limit jτj → 0. Signature change can occur when we go For this case we can apply coordinate transforma- away from either of these two limits, as we describe tion (4.22). Upon making the phase choice z3 ¼sinhτ, below. Eq. (5.13) can be written in the Taub-NUT form For the solutions given by (5.7) and (5.8),given (4.23), where the metric components are now some value of β˜ ð≠−1; −0.6Þ in the regions ði0Þ, ðii0Þ, and (iii), the sign of either gττ or gθθ changes at some 2 2 2 2 ˜ gττ ¼ 4ððμ þ ν − 2Þcosh τsinh τ − 1Þ; value of jτj. We plot the values of jτj versus β for 2 2 2 2 which this occurs in Fig. 4. gττ changes sign when gθθ ¼ cosh τðμ cosh τ − sinh τÞ; 2 2 2 2 ðμ þ ν − 2Þ sinh τ cosh τ ¼ 1 (indicated by the 2 2 gψψ ¼ −cosh τsinh τ: ð5:14Þ solid curves in Fig. 4). gθθ changes sign when

(a) (b) (c)

1;1 ˜ FIG. 4. Signature changes in the induced metric gμν and effective metric γμν for deformed CP are given in plots of jτj versus β in the 0 ˜ 0 ˜ ˜ three disconnected regions: (a) ði Þ − 3.414 ≲ β ≲ −0.746, (b) ðii Þ − 0.603 ≲ β ≲ −0.586, and (c) (iii) 13.67 ≲ β. A sign change in gθθ or γθθ is indicated by the dashed curves. A sign change in gττ or γψψ is indicated by the solid curves.

086015-10 SIGNATURE CHANGE IN MATRIX MODEL SOLUTIONS PHYS. REV. D 98, 086015 (2018)

tanh2 τ ¼ μ2 (indicated by the dashed curves in For the solutions given by (5.7) and (5.8), we find ˜ Fig. 4). Above the solid curves, signðgττ;gθθÞ¼ that for any fixed value of β in the regions ði0Þ, ðii0Þ, 0 ðþ; þÞ and det g< and so the induced metric has and (iii), either a sign change occurs for both gττ and a Lorentzian signature in this region. In this case, the gθθ, or there is no signature change. We plot the ψ θ ϕ 0;2 timelike direction corresponds to d þ cos d .It signature changes for deformed CP in Fig. 5. gθθ corresponds to a spacetime with closed timelike changes sign when coth2 τ ¼ μ2 (indicated by the curves. Above the dashed curves, signðgττ;gθθÞ¼ dashed curves in Fig. 5). gττ changes sign when ð−; −Þ, while det g>0. In this case, the induced 2 2 2 2 ðμ þ ν − 2Þ sinh τ cosh τ ¼ 1 (indicated by the metric space has a Euclidean signature. solid curves in Fig. 5). We find that there are no sign For the solution (5.11), a sign change in gθθ occurs ˜ 2 τ 2 changes in the induced metric for −1 < β ≲ −0.746 at tanh ¼ 5, and the induced metric has a Euclidean ˜ 2 2 and −0.603 ≲ β ≤−0.6. So for these subregions, the signature for tanh τ > 5. (2) Deformed CP0;2. signature of the induced metric remains Euclidean τ Here we apply the coordinate transformation for all . For the complementary subregions, a sign τ τ (4.26) to (5.13), along with the phase choice change occurs in gττ, say at j j¼j 1j, and gθθ,ata τ τ τ τ τ τ z3 ¼ cosh τ. The induced invariant interval again later j j, say j 2j, i.e., j 2j > j 1j.Forj j > j 2j, takes the Taub-NUT form (4.23), with the matrix signðgττ;gθθÞ¼ðþ; þÞ, and so the induced metric elements now being has a Lorentzian signature in this case. The timelike direction corresponds to dψ þ cos θdϕ, once again 2 2 2 2 gττ ¼ 4ððμ þ ν − 2Þcosh τsinh τ − 1Þ; corresponding to a spacetime with closed timelike 2 2 2 2 curves. For the intermediate interval in jτj where gθθ ¼ sinh τðμ sinh τ − cosh τÞ; jτ1j < jτj < jτ2j, we get signðgττ;gθθÞ¼ðþ; −Þ.In 2 2 gψψ ¼ −cosh τsinh τ; ð5:15Þ this case, the induced metric has a Lorentzian signature, with τ defining the timelike direction. 2 2 gϕϕ ¼ gψψ cos θ þ gθθ sin θ, and gψϕ ¼ gψψ cos θ. Any τ slice is topologically a three-sphere, since Only the results for gθθ differ in expressions (5.14) from (4.26), and (5.15). The latter reduce to that of undeformed 0;2 CP , Eq. (4.27), when μ ¼ ν ¼ 1. For that limit, as 2 2 2 jζ1j þjζ2j ¼ tanh τ: ð5:16Þ well as for jτj → 0, signðgττ;gθθÞ¼ð−; −Þ and det g>0 (away from coordinate singularities). In this case, the induced metric has a Euclidean Restricting to positive τ, the interval τ1 < τ < τ2 has 1 1 signature. As with deformed CP ; , signature change an initial singularity at τ1 and a final singularity at τ2. can occur when we go away from these limits, as we Therefore, although not very realistic, it describes a describe below. closed spacetime cosmology.

(a) (b) (c)

0;2 ˜ FIG. 5. Signature changes in the induced metric gμν and effective metric γμν for deformed CP are given in plots of jτj versus β for in the three disconnected regions: (a) ði0Þ − 3.414 ≲ β˜ ≲ −0.746, (b) ðii0Þ − 0.603 ≲ β˜ ≲ −0.586, and (c) (iii) 13.67 ≲ β˜. A sign change in gθθ or γθθ is indicated by the dashed curves. A sign change in gττ or γψψ is indicated by the solid curves. No sign changes occur in either the induced metric or the effective metric for −1 < β˜ ≲ −0.746 and −0.603 ≲ β˜ ≤−0.6.

086015-11 A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018)

No signature change in the induced metric results (indicated by the dashed curves in Fig. 4). The from the solution (5.11). effective metric has a Euclidean signature above the We remark that while the induced metrics for the two dashed curves. A sign change in γψψ (as with gττ) solutions 1. and 2. are modified from their undeformed appears when ðμ2 þ ν2 − 2Þ sinh2 τ cosh2 τ ¼ 1 (in- counterparts, their Poisson brackets, and corresponding dicated by the solid curves in Fig. 4). Above the solid symplectic two-forms, are unchanged. That is, for curves, the signature of the effective metric is Lor- deformed CP1;1 the symplectic two-form is (4.25) and entzian, det γ < 0, and τ is the timelike direction. A τ for deformed CP0;2 the symplectic two-form is (4.28). This slice again defines a three-sphere, since from (4.22), is relevant for the computation of the effective metric, ζ 2 ζ 2 2 τ which we do in the following subsection. j 1j þj 2j ¼ coth : ð5:19Þ Restricting to positive τ, this region with a Lorentzian 2. Effective metric signature has an initial singularity, and therefore, it describes an open spacetime. We shall see in the In Sec. IV we found that the induced metric gμν and the 1;1 0;2 next subsection that it corresponds to an expanding effective metric γμν for undeformed CP and CP are cosmology. identical. The same result does not hold for the corre- For the solution (5.11), a sign change in γθθ occurs sponding deformed solutions, as we show below. at tanh2 τ ¼ 2, and the effective metric has a Euclidean Furthermore, more realistic cosmologies follow from the 5 2 τ 2 effective metric of the deformed CP1;1 and CP0;2 solutions. signature for tanh > 5. There are no regions with a (1) Deformed CP1;1. To compute the effective metric we Lorentzian signature in this case. 0;2 need the symplectic matrix, as well as the induced (2) Deformed CP . We repeat the above calculation to γ 0;2 metric. For the deformed, as well as undeformed, get the effective metric μν for deformed CP . The CP1;1 solutions, the nonvanishing components of the inverse symplectic matrix is the same as for un- 0 2 inverse symplectic matrix are given in (B4), while deformed CP ; , with nonvanishing components 0 2 the induced metric for deformed CP1;1 is given by (B7). The induced metric for deformed CP ; is (5.14). In addition, j det Θj is given in (B5), while given in (5.15). Using the result for j det Θj in (B8), j det γj gets deformed, such that we now get

2 2 2 2 2 2 2 2 jdetγjjdetΘj¼4jgττjðμ cosh τ − sinh τÞ ; ð5:17Þ jdetγjjdetΘj¼4jgττjðμ sinh τ − cosh τÞ ; ð5:20Þ

with gττ given in (5.14). As a result, the nonvanish- with gττ given in (5.15). Now the nonvanishing ing components of the effective metric tensor are components of the effective metric tensor are found given by to be γ ττ γττ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ −1; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ −1; jdetγjjdetΘj j det γjj det Θj γθθ 1 γ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiθθ 4 μ2 − 2 τ ¼ 2 2 ; jdetγjjdetΘj ð tanh Þ j det γjj det Θj 4ðμ − coth τÞ γψψ 1 γ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ − ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiψψ 4 2τ 2τ 2 − μ2 − ν2 ¼ 2 2 2 ; jdetγjjdetΘj ðsech csch þ Þ j det γjj det Θj 4ðμ þ ν − 2 − 4csch 2τÞ ð5:18Þ ð5:21Þ

2 2 2 2 in addition to γϕϕ ¼ γψψ cos θ þ γθθ sin θ and again with γϕϕ ¼ γψψ cos θ þ γθθ sin θ and γψϕ ¼ γψϕ ¼ γψψ cos θ. The results again agree with the γψψ cos θ. The results reduce to the undeformed undeformed induced metric (4.24) in the μ ¼ ν ¼ 1 induced metric (4.27) in the limit μ ¼ ν ¼ 1, de- limit. This limit has two spacelike directions and two scribing a space with a Euclidean signature. timelike directions, with signðγψψ; γθθÞ¼ð−; þÞ For the solution given by (5.7) and (5.8), signature and det γ > 0 (away from coordinate singularities). changes in the effective metric occur at the same Signature changes occur in the effective metric for values of the parameters as the signature changes for the same values of the parameters at which the the induced metric, which are indicated in Fig. 5. γθθ 2 2 signature changes occur for the induced metric. (like gθθ) changes sign when coth τ ¼ μ (indicated For the solutions given by (5.7) and (5.8), by the dashed curves in Fig. 5). γψψ (like gττ) signature changes are again given in Fig. 4. A sign changes sign when ðμ2 þν2 −2Þsinh2 τcosh2 τ¼1 2 2 change in γθθ (as with gθθ) appears when tanh τ ¼ μ (indicated by the solid curves in Fig. 5). As seen

086015-12 SIGNATURE CHANGE IN MATRIX MODEL SOLUTIONS PHYS. REV. D 98, 086015 (2018)

in Fig. 5, given any fixed value of β˜ in the regions ði0Þ, Any time (τ) slice is a three-sphere, or more precisely, a 0 ðii Þ, and (iii), either a sign change occurs in both γψψ Berger sphere. These examples correspond to spacetime and γθθ or there is no signature change. No sign cosmologies with a big bang. To show that they are τ changes in the effective metric for −1 < β˜ ≲ −0.746 expanding we introduce a spatial scale aðj jÞ. We define τ and −0.603 ≲ β˜ ≤−0.6. So for these subregions it as the cubed root of the three-volume at any slice the signature of the induced metric remains Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Euclidean. For the complementary regions, a sign aðjτjÞ3 ¼ j det γð3Þjdθdϕdψ; ð5:22Þ 3 change occurs in γψψ at jτj¼jτ1j and γθθ at jτj¼jτ2j, S with jτ2j > jτ1j. In the intermediate region where γð3Þ denotes the effective metric on the τ slice. From jτ1j < jτj < jτ2j, signðγψψ; γθθÞ¼ðþ; −Þ.Herethe γð3Þ γ γ θ 2 effective metric has a Lorentzian signature, but unlike the form of the metric tensor, det ¼ ψψð θθ sin Þ , and γ γ τ what happens with the induced metric, dψ þ cos θdϕ since ψψ and θθ only depend on . Then is associated with the timelike direction, yielding qffiffiffiffiffiffiffiffiffiffi τ τ 3 2 closed timelike curves. For j j > j 2j, i.e., above the aðjτjÞ ¼ð4πÞ jγψψjjγθθj: ð5:23Þ dashed curves, signðγψψ; γθθÞ¼ðþ; þÞ, and so the effective metric picks up a Lorentzian signature, with τ τ We wish to determine how the spatial scale evolves with being the timelike direction. From (5.16),a slice is respect to the proper time t in the comoving frame a three-sphere. Restricting to positive τ, this region Z with a Lorentzian signature has an initial singularity, τ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 and so describes an open spacetime cosmology, which tðτÞ¼ −γττðτ Þdτ we next show, is expanding. Zτ0 τ No signature change in the effective metric results 1 1 ¼ j det γðτ0Þj4j det Θðτ0Þj4dτ0: ð5:24Þ from the solution (5.11). τ0 τ τ 3. Expansion The lower integration limit 0 corresponds to the value of at the big bang, i.e., the signature change. 1 1 0 2 From the deformed CP ; and CP ; solutions we found We next compute and plot aðjτjÞ versus tðτÞ for the two regions in parameter space where the effective metric has a cases, deformed CP1;1 and deformed CP0;2, in the regions Lorentzian signature, and possessed an initial singularity. of a Lorentzian signature:

(1) Deformed CP1;1. For the spatial volume, we get

3 2 3 2 2 2 1 2 2 2 2 1 aðjτjÞ ¼ð4πÞ cosh τj sinh τjjμ cosh τ − sinh τj2jðμ þ ν − 2Þ cosh τ sinh τ − 1j4; ð5:25Þ

after substituting (5.14) and (5.17) into (5.23). For the proper time tðjτjÞ in the comoving frame we get Z τ 2 2 2 1 2 2 2 2 1 tðτÞ¼2 jμ cosh τ0 − sinh τ0j2jðμ þ ν − 2Þ cosh τ0 sinh τ0 − 1j4dτ0; ð5:26Þ τ0

2 4 τ0 2τ0 2 2 τ and is associated with the signature change, given by sinh ¼ ðμ þν −2Þ. It corresponds to the value of at the initial singularity, where from (5.25), the spatial scale vanishes. In Fig. 6(a) we plot aðjτjÞ versus tðτÞ for regions of deformed CP1;1 where the effective metric has a Lorentzian signature, using three values of β˜. It shows a very rapid 1 5 expansion near the origin. For τ close to τ0, Eqs. (5.25) and (5.26) give a ∼ ðτ − τ0Þ12 and t ∼ ðτ − τ0Þ4. Hence, 1 a ∼ t15. For large τ, a is linear in t. The same large distance behavior was found for solutions in [13]. (2) For deformed CP0;2, Eq. (5.23) gives

3 2 3 2 2 2 1 2 2 2 2 1 aðjτjÞ ¼ð4πÞ j sinh τj cosh τjμ sinh τ − cosh τj2jðμ þ ν − 2Þ cosh τ sinh τ − 1j4; ð5:27Þ

after using (5.15) and from (5.20). Equation (5.24) gives Z τ 2 2 2 1 2 2 2 2 1 tðτÞ¼2 jμ sinh τ0 − cosh τ0j2jðμ þ ν − 2Þ cosh τ0 sinh τ0 − 1j4dτ0: ð5:28Þ τ0

086015-13 A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018) 1 2 ˜ 7 10 ð2μ − αÞ μ − þ 3μβ ¼ 0; 6 2 8 5 μ2 ν2 − 2 − α μ ν 4β˜ 0 4 6 þ ð þ Þþ ¼ ; 3 ˜ 4 2νðβ − 1Þþα ¼ 0; ð5:30Þ 2 2 1 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 which differs from (5.6) in various signs. We can ˜ (a) (b) obtain (5.30) by making the replacement ðα; β; μ; νÞ → ðiα; −β˜;iμ;iνÞ in (5.6). To obtain a solution to (5.30),we FIG. 6. aðjτjÞ versus tðτÞ for regions of (a) deformed CP1;1 and can then make the same replacement in the solution (5.7). (b) deformed CP0;2 where the effective metric has a Lorentzian The result is signature (and τ is the timelike direction) for β˜ ¼ −3 (solid curve), −0.595 (dashed curve), and 14 (dot-dashed curve). β˜ 2 þ β˜ − 1 − γ½−β˜ α ¼ 2μ ; −2β˜ þ 1 τ τ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Again, the initial value 0 for is associated with a ˜ 3 ˜ 2 ˜ ˜ ˜ 2 2 β þ 4β − 6β þ βγ½−βþ2 signature change, now satisfying coth τ0 ¼ μ .It μ ¼ ˜ 2 ˜ ; corresponds to a big-bang singularity, and from 2ðβ − 4β þ 2Þ (5.27), aðjτ0jÞ ¼ 0. In Fig. 6(b) we plot aðjτjÞ versus α 0;2 ν ¼ ; ð5:31Þ tðτÞ for regions of deformed CP where the 2ð1 − β˜Þ effective metric has a Lorentzian signature, using three values of β˜. It too shows a rapid expansion near γ β˜ μ ν α the origin. For τ close to τ0, Eqs. (5.27) and (5.28) where ½ was defined in (5.8). The parameters , , and 1 3 1 γ −β˜ a ∼ τ − τ0 6 t ∼ τ − τ0 2 a ∼ t9 (and necessarily, ½ ) are all real only for the following give ð Þ and ð Þ . Hence, . ˜ As with the case of deformed CP1;1 and [13], a ∼ t two disconnected intervals in β: for large τ. ðiÞ 0.325 ≲ β˜ ≲ 0.586; ðiiÞ 3.41 ≲ β˜: ð5:32Þ B. Deformed CP2 We now look for solutions to the previous eight- dimensional matrix model that are deformations of noncommutative CP2. CP2 is a solution to an eight- 1. Induced metric dimensional matrix model in a Euclidean background. In The metric induced from the flat background metric (4.9) [12], such solutions were found when the background onto the surface spanned by (5.29) is metric is changed to diagðþþþþþþþ−Þ. Here we show that deformations of noncommutative CP2 solve the matrix model with the indefinite metric (4.9), and that they ds2 ¼ dxadx a may be associated with multiple signature changes. 1 We once again assume that the three complex coordi- −ds2 1 dx2 dx2 dx2 ¼ FS þ þ μ2 ð 1 þ 2 þ 3Þ nates zi satisfy the constraint (4.4), but now that the indices are raised and lowered with the three-dimensional 1 1 dx2; : Euclidean metric. The Poisson brackets that arise from þ þ ν2 8 ð5 33Þ the commutative limit of fuzzy CP2 are (4.6) (now, assuming the Euclidean metric) [12]. We replace the 2 ˜ where dsFS denotes the Fubini-Study metric suð2; 1Þ Gell-Mann matrices λa in (5.5) by suð3Þ Gell- Mann matrices λa, i.e., X8 μ λ i j λ i j 2 †λ 2 4 2 − † 2 x1–3 ¼ zi ½ 1−3 jz ;x4–7 ¼ zi ½ 4−7 jz ; dsFS ¼ ðdðz azÞÞ ¼ ðjdzj jz dzj Þ: ð5:34Þ 1 ν λ i j a¼ x8 ¼ zi ½ 8 jz : ð5:29Þ

2 i Now substitute this ansatz into the equations of motion Here we introduce the notation jdzj ¼ dzi dz , † i †λ λ i j (5.3) to get the following conditions on the parameters: z dz ¼ zi dz , and z az ¼ zi ½ a jz . Then (5.33) becomes

086015-14 SIGNATURE CHANGE IN MATRIX MODEL SOLUTIONS PHYS. REV. D 98, 086015 (2018)

ds2 ¼ −4ðjdzj2 − jz†dzj2Þ 2 2 2 2 2 2 2 þ 4ðμ þ 1Þðjz1j þjz2j Þðjdz1j þjdz2j Þþðμ þ 1Þðz1dz1 þ z2dz2 − z1dz1 − z2dz2Þ 1 ν2 1 − 2 − 2 2 þ 3 ð þ Þðz1dz1 þ z2dz2 z3dz3 þ z1dz1 þ z2dz2 z3dz3Þ : ð5:35Þ

2 2 3 −2 Next introduce local coordinates ðζ1; ζ2Þ defined in (4.18), now satisfying jζ1j þjζ2j þ 1 ¼jz j . Then

2 2 2 2 2 2 ds ¼ −4ðz3Þ ðjdζ1j þjdζ2j − ðz3Þ jΞj Þ 1 4 μ2 1 1 − 2 2 − 1 2 ζ 2 ζ 2 Ξ Ξ þ ð þ Þð ðz3Þ Þ ðdz3Þ 2 þðz3Þ ðjd 1j þjd 2j Þþz3dz3ð þ Þ ðz3Þ 2 2 2 2 2 þðμ þ 1Þððz3Þ ðΞ − Ξ ÞÞ − ðν þ 1Þð2z3dz3Þ ; ð5:36Þ

where we again chose z3 to be real and defined intermediate region between the dashed and dot-dashed Ξ ¼ ζ1dζ1 þ ζ2dζ2. We introduce Euler-like angles curves, with dψ þ cos θdϕ timelike. Above the dot-dashed ðθ; ϕ; ψÞ, along with τ, which now is an angular variable, curve, the induced metric again has two spacelike direc- π 0 ≤ τ < 2, using tions and two timelike directions. θ i ψ ϕ ζ 2ð þ Þ τ 1 ¼ e cos 2 tan ; 2. Effective metric θ iðψ−ϕÞ We next use (4.29) to compute the effective metric γμν for ζ2 ¼ e2 sin tan τ: ð5:37Þ 2 deformed CP2. Starting with the canonical Poisson brack- 2 2 It then follows that ðz3Þ ¼ cos τ. The induced invariant ets (4.6), we now obtain the following results for the interval again takes the Taub-NUT form (4.23), with the nonvanishing components of the symplectic matrix ½Θμν: nonvanishing matrix elements 1 2cotθ 2cscθ 4 −1 μ2 − ν2 2τ 2τ Θτψ ¼ ; Θθψ ¼ ; Θθϕ ¼ − : gττ ¼ ð þð Þsin cos Þ; sinτcosτ sin2τ sin2τ 2 2 2 2 gθθ ¼ sin τð−cos τ þ μ sin τÞ; ð5:39Þ 2 2 gψψ ¼ −sin τcos τ; ð5:38Þ Computing determinants, we get 2 2 along with gϕϕ ¼ gψψ cos θ þ gθθ sin θ and gψϕ ¼ gψψ cos θ. The induced metric has a Euclidean signature for τ close to zero. A sign change in gθθ occurs τ 1 μ2 − ν2 1 for tan ¼ jμj.If > 4, two additional signature changes occur in the induced metric for the domain π 0 < τ < 2. Specifically, gττ changes sign when sin 2τ ¼ pffiffiffiffiffiffiffiffiffi2 . We find numerically that μ2 < ν2 for μ2−ν2 solutions (5.31) with β˜ in the region (i) in (5.32), and that μ2 > ν2 in the region (ii). So only one signature change occurs when β˜ has the values in (i). It is a change from the Euclidean signature to one where the induced metric has two spacelike directions and two timelike directions. On the other hand, three signature changes can occur when β˜ has values in (ii). They are plotted as a function of β˜ in Fig. 7. A sign change in gθθ is indicated by the solid curve, and sign changes in gττ are indicated by the dashed and dot-dashed curves. The induced metric has a Euclidean FIG. 7. Signature changes in the induced metric gμν and 2 signature below the solid curve. In the tiny intermediate effective metric γμν for deformed CP are given in the plot of ˜ ˜ region between the dashed and the solid curves, the induced τ versus β in the region (ii) 3.41 ≲ β. A sign change in gθθ or γθθ is metric has two spacelike directions and two timelike indicated by the solid curve. Sign changes in gττ or γψψ are directions. It has a Lorentzian signature in the other indicated by the dashed and dot-dashed curves.

086015-15 A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018)

4 csc2 θ det Θ ¼ ; cos2 τ sin6 τ 12 2 2 2 2 10 j det γjj det Θj¼jdet gjj det Θ¼j4jgττjðcos τ − μ sin τÞ : 8 ð5:40Þ 6 As a result, the nonvanishing components of the effective 4 metric tensor are 2

γττ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ −1; 10 20 30 40 50 60 70 j det γjj det Θj τ τ 2 γ 1 FIG. 8. að Þ versus tð Þ for the region of deformed CP where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiθθ β˜ 3 5 ¼ 2 2 ; the effective metric has a Lorentzian signature, for ¼ . (solid j det γjj det Θj 4ðμ − cot τÞ curve), 3.75 (dashed curve), and 5 (dot-dashed curve). γ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiψψ ¼ 2 2 2 2 ; ð5:41Þ has the values in (ii). In this case, τ is the timelike j det γjj det Θj 4ðμ − ν − sec τ csc τÞ coordinate, and it evolves from one signature change to 2 2 2 2 2 another. From (5.37), jζ1j þjζ2j ¼ tan τ, and so, as with in addition to γϕϕ ¼ γψψ cos θ þ γθθ sin θ and γψϕ ¼ 1;1 0;2 τ 1;1 0;2 the deformed CP and CP solutions, a slice of the γψψ cos θ. As with the deformed CP and CP solutions, four-dimensional manifold away from singularities is a signature changes in the effective metric coincide with three-sphere, or more precisely, a Berger sphere. We can signature changes in the induced metric. So as with the compute the spatial scale aðjτjÞ at any τ slice and proper induced metric, the effective metric undergoes only one 2 time t in the comoving frame for the deformed CP β˜ signature change when has the values in (i). It is a change solution, using (5.23) and (5.24), respectively. For the from the Euclidean signature to onewhere the effective metric former, we get has two spacelike directions and two timelike directions. Also as with the induced metric, the effective metric 3 2 4 2 2 1 2 2 aðjτjÞ ¼ð4πÞ j sin τ cos τjj cot τ − μ j2jðμ − ν Þ undergoes three signature changes when β˜ has the values in 2 2 1 (ii), which are indicated in Fig. 7. A sign change in γθθ × sin τ cos τ − 1j4: ð5:42Þ τ 1 occurs for tan ¼ jμj (indicated by the solid curve in Fig. 7), 2 It follows that the spatial scale vanishes at the signature and sign changes in γψψ occur at sin 2τ ¼ pffiffiffiffiffiffiffiffiffi (indicated μ2−ν2 changes, which are associated with the cosmological by the dashed and dot-dashed curves in Fig. 7). The singularities. For the latter, Eq. (5.24) gives effective metric has a Euclidean signature below the solid Z τ curve. In the tiny intermediate region between the solid and 2 2 2 1 2 2 tðτÞ¼2 dτ0j cos τ0 − μ sin τ0j2jðμ − ν Þ the dashed curves, the effective metric, like the induced τ0 metric, has two spacelike directions and two timelike 2 2 1 τ0 τ0 − 1 4 directions. The effective metric has a Lorentzian signature × sin cos j : ð5:43Þ in the intermediate region between the dashed and dot- τ The lower integration limit τ0 corresponds to the value of τ dashed curves, with being timelike. Above the dot-dashed 2 at the coordinate singularity defined by sin 2τ0 ¼ pffiffiffiffiffiffiffiffiffi, curve, the induced metric has two spacelike directions and μ2−ν2 two timelike directions. π τ0 < 4, corresponding to the sign change in γψψ. In Fig. 8, For the Lorentzian region, which we found between the we plot aðτÞ versus tðτÞ for three values of β˜ in region (ii). dashed and dot-dashed curves in Fig. 7, the effective metric 1 5 β˜ For τ close to τ0, we get a ∼ ðτ − τ0Þ12, t ∼ ðτ − τ0Þ4, and describes a closed spacetime cosmology. For a fixed with 1 ∼ 15 values in (ii), the sign changes in γψψ, depicted as red and hence, a t . We find identical behavior near the other τ τ blue curves in Fig. 7, correspond to spacetime singularities. singularity at ¼ 1. We thus get a very rapid initial We denote the values of τ at these singularities by τ0 and τ1, expansion and a very rapid final contraction. with τ0 < τ1. Which one of these is the initial singularity, and which one is the final singularity, of course, depends on VI. CONCLUSIONS the direction of time. We obtain the time evolution of the We have obtained a number of new solutions to IKKT-type spatial scale for this region in the next subsection. matrix models, which exhibit signature change in the commutative limit. All such examples found so far, including 3. Expansion and contraction the noncommutative H4 solution of [13], require including a In the previous section, we saw that the effective metric mass term in the matrix action. Since mass terms result from for deformed CP2 can have a Lorentzian signature when β˜ an IR regularization [16], it is interesting to speculatewhether

086015-16 SIGNATURE CHANGE IN MATRIX MODEL SOLUTIONS PHYS. REV. D 98, 086015 (2018) signature change on the brane is connected to the regulari- limit, one recovers the CP1;2 manifold, an S2 bundle zation.5 On the other hand, we remark that the mass term over AdS4. resulting from the regularization does not necessarily lead to The eight-dimensional matrix model considered in a signature change, since we have obtained solutions to the Secs. IV and V utilized a particular indefinite background massive matrix model that exhibit no signature change in the metric η, the suð2; 1Þ Cartan-Killing metric. Other indefinite commutative limit. For example, no sign changes occur in background metrics can be considered. η ¼ diagðþ þ þ þ either the induced metric or the effective metric for deformed þþþ−Þ was used in [12], to obtain noncommutative CP2 CP0;2 when −1 < β˜ ≲ −0.746 and −0.603 ≲ β˜ ≤−0.6. solutions. We can preserve SOð3Þ rotational symmetry Moreover, our work does not rule out the possibility of with a generalization of the background metric to solutions to a massless matrix model that exhibit signature η ¼ diagðκ3; κ3; κ3; −; −; −; −; κ8Þ, κ3, κ8 ¼. (We exclude change in the commutative limit. κ3 ¼ κ8 ¼ −, since this will only produce a Euclidean The four-dimensional solutions of Sec. V are deforma- induced and effective metric.) If, e.g., we search for deformed tions of noncommutative complex projective spaces, spe- CP1;1 and CP0;2 solutions to the matrix equations (5.3) with cifically noncommutative CP2, CP1;1, and CP0;2. The this metric, then the conditions (5.6) generalize to manifolds that emerge from these solutions can have multiple 1 signature changes. The manifolds resulting from a deformed 2 ˜ ð2μ − αÞ μ κ3 þ þ 3μβ ¼ 0; noncommutative CP0;2 solution can undergo two signature 2 changes, while those resulting from a deformed noncommu- 2 2 ˜ μ κ3 þ ν κ8 þ 2 − αðμκ3 þ νκ8Þþ4β ¼ 0; tative CP2 solution can have up to three signature changes. 2ν − α 2νβ˜ 0 The regions where the effective metric of these manifolds þ ¼ ; ð6:1Þ have a Lorentzian signature serve as crude models of closed where we again assumed the ansatz (5.5). Solutions for (in the case of noncommutative CP2) and open (in the case of different choices of κ3 and κ8 may be found, although they 1;1 0;2 noncommutative CP and CP ) cosmological spacetimes. may be quite nontrivial, and many more four-dimensional They contain cosmological singularities that are resolved signature changing manifolds are expected to emerge in the away from the commutative limit. The evolution of the commutative limit. spatial scale a as a function of the proper time t in the In this article we have neglected stability issues and the comoving frame was computed for these examples. For all addition of fermions. The question of stable solutions to 4 examples (and also the example of noncommutative H in matrix models is highly nontrivial. For two-dimensional [13]) an extremely rapid expansion (or contraction, in the solutions it was found previously that longitudinal and case of the big crunch singularity of the closed cosmology) transverse fluctuations contribute with opposite signs to the was found for the spatial scale a near the cosmological kinetic energy. It is unclear how the extension to a fully singularities. Rather than following an exponential behavior, supersymmetric theory can resolve this issue. We hope to 1 2 we obtained a ∼ t15 near t ¼ 0 for noncommutative CP and address such questions in the future. 1 1 1 0 2 CP ; ,anda ∼ t9 for noncommutative CP ; . Also like noncommutative H4 [13], the spacetimes emerging from ACKNOWLEDGMENTS 1 1 0 2 the deformed noncommutative CP ; and CP ; solutions A. S. is grateful for useful discussions with J. Hoppe, H. expand linearly at late times, a ∼ t. Kawai, S. Kurkcuoglu, and H. Steinacker, and thanks the Unlike the spacetime manifold that emerges from non- members of the Erwin Schrödinger International Institute for 4 commutative H [13], the manifolds that emerge from their hospitality and support during the Workshop on Matrix 2 1 1 0 2 noncommutative CP , CP ; , and CP ; are not maximally Models for Noncommutative Geometry and String Theory. symmetric. For the latter manifolds, any time slice of the spacetime is a Berger sphere. Although being, perhaps, less APPENDIX A: SOME PROPERTIES OF suð2;1Þ IN realistic than noncommutative H4 with regards to cosmol- THE DEFINING REPRESENTATION ogy, the examples of noncommutative CP2, CP1;1, and In terms of suð3Þ Gell-Mann matrices λ , the suð2; 1Þ CP0;2 are considerably simpler spaces than noncommuta- a Gell-Mann matrices λ˜ are given by tive H4, with evidently similar outcomes for the evolution a of the spatial scale. Noncommutative H4 carries an addi- λ˜ ¼ λ ; a ¼ 1; 2; 3; 8; tional bundle structure that is not present for the solutions a a λ˜ λ 0 4 5 6 7 of Sec. V. In order to close the algebra on noncommutative a0 ¼ i a0 ; a ¼ ; ; ; : ðA1Þ H4, one must extend it to a larger noncommutative space. That space is noncommutative CP1;2. In the commutative They satisfy the Hermiticity properties (4.7). c ˜ dc The structure constants for suð2; 1Þ are Cab ¼ fabdη , ˜ where ηab is the Cartan-Killing metric (4.9), and fabc are 5We thank the referee for this remark. totally antisymmetric, with the nonvanishing values

086015-17 A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018) pffiffiffi 3 by iℏ times Poisson brackets. We also need to replace ˜ 1 ˜ ˜ − R f123 ¼ ; f845 ¼ f867 ¼ 2 ; the trace by an integration dμðxÞ, where dμðxÞ is an 1 invariant integration measure. Say that the manifold is ˜ ˜ ˜ ˜ ˜ ˜ − 1 2 f147 ¼ f165 ¼ f246 ¼ f257 ¼ f345 ¼ f376 ¼ 2 : ðA2Þ parametrized by σ ¼ðσ ; σ ; …; σnÞ, n ≤ d, with symplec- 1 −1 μ ν tic two-form Ω ¼ 2 ½Θ μνdσ ∧ dσ . Then one can set ˜ σ Except for f123 these structure constants are opposite in dμðxÞ¼pffiffiffiffiffiffiffiffiffiffid . Taking k → ℏκ, the semiclassical limit of sign from those obtained from the standard Gell-Mann j det Θj matrices of suð3Þ. (B1) is 2 1 Z Some useful identities for the suð ; Þ Gell-Mann 1 dσ ˜ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ a ϕ matrices and f are 2 fxa; gfx ; g abc 2κ j det Θj ˜ ˜ ˜ ˜ Z trλ λ ¼½λ i ½λ j ¼ 2η ; ðA3Þ 1 dσ a b a j b i ab − pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Θρμ∂ ∂ ϕΘσν∂ a∂ ϕ ¼ 2 ρxa μ σx ν 4 2κ j det Θj λ˜ λ˜ 2˜ λ˜c η 1 Z ½ a; bþ ¼ dabc þ 3 ab ; ðA4Þ 1 dσ ρμ σν − pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Θ gρσΘ ∂μϕ∂νϕ: B2 ˜ ˜ bc 3η ¼ 2κ2 ð Þ fabcf d ¼ ad; ðA5Þ j det Θj 2 λã i λ˜ k 2δi δk − δi δk On the other hand, the standard action of a scalar field ϕ on ½ j½ a l ¼ l j 3 j l: ðA6Þ a background metric γμν is Z ½; denotes the anticommutator, and d˜ are totally pffiffiffiffiffiffiffiffiffiffiffiffiffi þ abc 1 μν − dσ γ γ ∂μϕ∂νϕ: symmetric, with the nonvanishing values 2κ2 j det j ðB3Þ ˜ ˜ ˜ ˜ ˜ 1 d443 ¼ d553 ¼ d146 ¼ d157 ¼ d256 ¼ − ; Identifying these two actions gives (4.29). 2 As examples, we compute the effective metrics for ˜ ˜ ˜ 1 (undeformed) CP1;1 and CP0;2, and show that they are d663 ¼ d773 ¼ d247 ¼ ; 2 identical to the corresponding induced metrics. 1 1 ˜ ˜ ˜ 1 (1) Effective metric for CP ; . Using (4.25), the non- d118 ¼ d228 ¼ d338 ¼ pffiffiffi ; μν 1 1 3 vanishing components Θ for CP ; are 1 ˜ ˜ ˜ ˜ 1 2 cot θ d448 ¼ d558 ¼ d668 ¼ d778 ¼ pffiffiffi ; Θτψ Θθψ 2 3 ¼ ; ¼ 2 ; cosh τ sinh τ cosh τ 1 2 θ ˜ ffiffiffi θϕ csc d888 ¼ − p : ðA7Þ Θ ¼ − : ðB4Þ 3 cosh2τ Equation (A6) is the Fierz identity, which has the same Then form as that for suð3Þ. 4 2θ Θ csc det ¼ 2 6 ; APPENDIX B: EFFECTIVE METRIC sinh τcosh τ j det γj¼4cosh6τsinh2τsin2θ: ðB5Þ Here we review the derivation of (4.29), relating the γ effective metric μν to the induced metric. We use the Computing ΘTgΘ we find the following nonvanish- example of the massless scalar field [15]. We then use the ing components: result to compute the effective metrics for (undeformed) 1 1 0 2 CP ; and CP ; . ½ΘTgΘττ ¼ −1; Φ Φ Denote the scalar field by ¼ ðXÞ on a noncommu- 4 tative background spanned by matrices X . The standard ΘTgΘ θθ ; a ½ ¼ 2τ action is cosh 4csc2θ ΘT Θ ϕϕ 1 ½ g ¼ 2 ; − X ; Φ XaΦ ; cosh τ 2 2 Tr½ a ½ ðB1Þ k 4ðcot2θ − csch2τÞ ½ΘTgΘψψ ¼ ; a ¼ 1; 2; …;d. Now take the semiclassical limit ℏ → 0. cosh2τ This means again replacing matrices Xa by commuting 4 cot θ csc θ ½ΘTgΘϕψ ¼ − : ðB6Þ variables xa, corresponding to embedding coordinates of cosh2τ some continuous manifold. Φ is then replaced by a function ϕ on the manifold, and commutators are replaced Using (4.29) and (B5), we then get γμν ¼ gμν.

086015-18 SIGNATURE CHANGE IN MATRIX MODEL SOLUTIONS PHYS. REV. D 98, 086015 (2018)

0 2 (ii) Effective metric for CP ; . Using (4.28) the non- ½ΘTgΘττ ¼ −1; μν 0 2 vanishing components Θ for CP ; are −4 ΘTgΘ θθ ; ½ ¼ 2τ 1 2cotθ 2cscθ sinh Θτψ Θθψ Θθϕ − 2 ¼ ; ¼ 2 ; ¼ 2 : −4csc θ coshτsinhτ sinh τ sinh τ ΘT Θ ϕϕ ½ g ¼ 2τ ; ðB7Þ sinh −4ðcot2θ þ sech2τÞ ½ΘTgΘψψ ¼ ; Here sinh2τ ϕψ 4 cot θ csc θ 4 2θ ½ΘTgΘ ¼ : ðB9Þ csc 2 6 2 2τ detΘ¼ ; jdetγj¼4cosh τsinh τsin θ: sinh sinh6τcosh2τ ðB8Þ γ The nonvanishing components of ΘTgΘ are Using (4.29) and (B8), we once again get μν ¼ gμν.

4 [1] D. Atkatz and H. Pagels, The origin of the Universe as a 4-hyperboloid Hn and higher-spin in Yang-Mills matrix quantum tunneling event, Phys. Rev. D 25, 2065 (1982). models, arXiv:1806.05907. [2] A. Vilenkin, Creation of universes from nothing, Phys. Lett. [14] N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, A 117B, 25 (1982). large N reduced model as superstring, Nucl. Phys. B498, [3] A. D. Sakharov, Cosmological transitions with a change in 467 (1997). metric signature, Zh. Eksp. Teor. Fiz. 87, 375 (1984) [15] For an introduction see H. Steinacker, Emergent geometry [Sov. Phys. JETP 60, 214 (1984)]; [Sov. Phys. Usp. 34, and gravity from matrix models: An introduction, Classical 409 (1991)]. Quantum Gravity 27, 133001 (2010). [4] G. W. Gibbons and J. B. Hartle, Real tunneling geometries [16] S. W. Kim, J. Nishimura, and A. Tsuchiya, Expanding and the large scale topology of the Universe, Phys. Rev. D (3 þ 1)-Dimensional Universe from a Lorentzian Matrix 42, 2458 (1990). Model for Superstring Theory in (9 þ 1)-Dimensions, Phys. [5] T. Dray, C. A. Manogue, and R. W. Tucker, Particle pro- Rev. Lett. 108, 011601 (2012); Expanding universe as a duction from signature change, Gen. Relativ. Gravit. 23, 967 classical solution in the Lorentzian matrix model for non- (1991). perturbative superstring theory, Phys. Rev. D 86, 027901 [6] M. Mars, J. M. M. Senovilla, and R. Vera, Signature Change (2012). on the Brane, Phys. Rev. Lett. 86, 4219 (2001). [17] C. Gu, The extremal surfaces in 3-dimensional Minkowski [7] J. Mielczarek, Signature change in loop quantum cosmol- space, Acta Math. Sin. 1, 173 (1985). ogy, Springer Proc. Phys. 157, 555 (2014); J. Mielczarek [18] V. A. Klyachin, Zero mean curvature surfaces of mixed type and T. Trzeniewski, Spectral dimension with deformed in Minkowski space, Izv. Math. 67, 209 (2003). spacetime signature, Phys. Rev. D 96, 024012 (2017). [19] Y. W. Kim and S.-D. Yang, Prescribing singularities of [8] J. Ambjorn, D. N. Coumbe, J. Gizbert-Studnicki, and J. maximal surfaces via a singular Björling representation Jurkiewicz, Signature change of the metric in CDT quantum formula, J. Geom. Phys. 57, 2167 (2007). gravity?, J. High Energy Phys. 08 (2015) 033. [20] Y. W. Kim, S.-E. Koh, H. Shin, and S.-D. Yang, Spacelike [9] A. Barrau and J. Grain, Cosmology without time: What to maximal surfaces, timelike minimal surfaces, and Björling do with a possible signature change from quantum gravi- representation formulae, J. Korean Math. Soc. 48, 1083 tational origin?, arXiv:1607.07589. (2011). [10] M. Bojowald and S. Brahma, Signature change in loop [21] S. Fujimori, Y. W. Kim, S.-E. Koh, W. Rossman, H. Shin, quantum gravity: General midisuperspace models and M. Umehara, K. Yamada, and S.-D. Yang, Zero mean dilaton gravity, Phys. Rev. D 95, 124014 (2017). curvature surfaces in Lorentz–Minkowski 3-space which [11] A. Chaney, L. Lu, and A. Stern, Lorentzian fuzzy spheres, change type across a light-like line, Osaka J. Math. 52, 285 Phys. Rev. D 92, 064021 (2015). (2015). [12] A. Chaney and A. Stern, Fuzzy CP2 spacetimes, Phys. Rev. [22] P. M. Ho and M. Li, Large N expansion from fuzzy AdS(2), D 95, 046001 (2017). Nucl. Phys. B590, 198 (2000); Fuzzy spheres in AdS/CFT [13] H. C. Steinacker, Cosmological space-times with resolved correspondence and holography from noncommutativity, big bang in Yang-Mills matrix models, J. High Energy Phys. Nucl. Phys. B596, 259 (2001). 02 (2018) 033; Quantized open FRW cosmology from [23] D. Jurman and H. Steinacker, 2D fuzzy anti-de Sitter Yang-Mills matrix models, Phys. Lett. B 782, 176 space from matrix models, J. High Energy Phys. 01 (2018); M. Sperling and H. C. Steinacker, The fuzzy (2014) 100.

086015-19 A. STERN and CHUANG XU PHYS. REV. D 98, 086015 (2018)

[24] A. Stern, Matrix model cosmology in two space-time [32] A.P. Balachandran, B.P. Dolan, J.H. Lee, X. dimensions, Phys. Rev. D 90, 124056 (2014). Martin, and D. O’Connor, Fuzzy complex projective [25] A. Chaney, L. Lu, and A. Stern, Matrix model approach to spaces and their star products, J. Geom. Phys. 43,184 cosmology, Phys. Rev. D 93, 064074 (2016). (2002). [26] A. Pinzul and A. Stern, Noncommutative AdS2=CFT1 [33] T. Azuma, S. Bal, K. Nagao, and J. Nishimura, Dynamical duality: The case of massless scalar fields, Phys. Rev. D aspects of the fuzzy CP2 in the large N reduced model with 96, 066019 (2017). a cubic term, J. High Energy Phys. 05 (2006) 061. [27] M. Chaichian, A. Demichev, P. Presnajder, and A. Tureanu, [34] D. Karabali, V. P. Nair, and S. Randjbar-Daemi, Fuzzy Space-time noncommutativity, discreteness of time and spaces, the M(atrix) model and the quantum Hall effect, unitarity, Eur. Phys. J. C 20, 767 (2001); Non-commutative in From Fields to Strings, edited by M. Shifman quantum field theory: Unitarity and discrete time, Phys. et al. (World Scientific, Singapore, 2005), Vol. 1, Lett. B 515, 426 (2001). pp. 831–875. [28] A. P. Balachandran, T. R. Govindarajan, A. G. Martins, and [35] H. Grosse and H. Steinacker, Finite gauge theory on fuzzy 2 P. Teotonio-Sobrinho, Time-space noncommutativity: CP , Nucl. Phys. B707, 145 (2005). Quantised evolutions, J. High Energy Phys. 11 (2004) 068. [36] A. P. Balachandran, S. Kurkcuoglu, and S. Vaidya, Lectures [29] A. Stern, Noncommutative static strings from matrix mod- on Fuzzy and Fuzzy SUSY Physics, (World Scientific, els, Phys. Rev. D 89, 104051 (2014). Singapore, 2007). [30] V. P. Nair and S. Randjbar-Daemi, On brane solutions in [37] K. Hasebe, Non-compact Hopf maps and fuzzy ultra- M(atrix) theory, Nucl. Phys. B533, 333 (1998). hyperboloids, Nucl. Phys. B865, 148 (2012). [31] G. Alexanian, A. P. Balachandran, G. Immirzi, and B. Ydri, [38] J. Arnlind and J. Hoppe, The world as quantized minimal Fuzzy CP2, J. Geom. Phys. 42, 28 (2002). surfaces, Phys. Lett. B 723, 397 (2013).

086015-20