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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 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;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 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].
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