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Hep-Th] 19 Jun 2008 Pc Ie( Ult Soeo H Aoseape.We Example) MIFP-07 Supercriticality of a Class of Critical String Cosmological Solutions Dimitri V. Nanopoulos1,2,3 and Dan Xie1 1George P. and Cynthia W.Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843, USA 2Astroparticle physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, TX 77381, USA 3Academy of Athens, Division of Nature Sciences, 28 panepistimiou Avenue, Athens 10679, Greece (Dated: May 26, 2018) For a class of Friedmann-Robertson-Walker-type string solutions with compact hyperbolic spatial slices formulated in the critical dimension, we find the world sheet conformal field theory which involves the linear dilaton and Wess-Zumino-Witten-type model with a compact hyperbolic target space. By analyzing the infrared spectrum, we conclude that the theory is actually supercritical due to modular invariance. Thus, taking into account previous results, we conclude that all simple non- trivial string cosmological solutions are in fact supercritical. In addition, we discuss the relationship of this background with the Supercritical String Cosmology (SSC). PACS numbers: I. INTRODUCTION expect that strings experience the space-time dimensions in a novel way, and it may be possible that string theory String theory is usually formulated in the critical di- formulated in critical dimensions actually exhibits non- mension (D=26 for the bosonic string, and D=10 for the critical behavior, i.e. the number of effective space time dimensions in which strings can freely oscillate is larger superstring ), and there are various ways to justify why this is the natural choice[1]. The idea of a critical dimen- than the critical dimension. sion has been firmly entrenched in the way that string In [6], the authors analyzed the conformal field the- theory is usually formulated, and it is widely taken for ory of FRW type solutions with space curvature κ = 0 granted that this should be the case in most studies. Nev- and κ = +1; it was shown that the space dimensions ertheless, it is possible to formulate string theory in a should be supercritical. In this paper, we will consider dimension other than the critical dimension, and such a the case with space curvature κ = 1. This solution did way of formulating string theory finds a useful applica- not attract much attention because− conformal field the- tion in studying time-varying backgrounds such as those ory with hyperbolic target space is hard to study. For which are necesssary for cosmology [2][3]. this solution, we can choose the time-like linear dilaton There are several reasons to suspect that non-critical and the level of the hyperbolic CFT properly so that the string theory is relevant to the web of string duality. The space time dimensions are critical. This solution is triv- first clue comes from Cosmology. Observational data ial if the hyperbolic manifold is noncompact because the from the SnIa [4] project and WMAP 1, 3 [5] strongly background is only part of the flat Minkowski space. The suggests that the universe is currently in an accelerating amazing thing is that if we make the hyperbolic manifold phase, which may be explained by the presence of dark compact, (to make the solution topologically nontrivial) energy. String theory, as a theory of quantum gravity, the actual space time dimensions of this background is should give an explanation of the dark energy. However, supercritical. So we conclude for all the nontrivial FRW it is very difficult to find exact time-dependent solutions arXiv:0710.2312v2 [hep-th] 19 Jun 2008 type solutions, the theory should be formulated in non- in critical string theory. However, it has been shown in critical dimensions. [6][7] that if one goes beyond the critical dimension, one can find simple time-dependent Friedmann-Robertson- This paper is organized in the following way: In section Walker (FRW) solutions with a time-like linear dilaton. II, we discuss the cosmological solutions derived from the So-called supercritical strings (for strings with dimension string equations of motion and the quantization of the su- D>Dcritical) may then also provide an explanation for percritical linear dilaton background. It is shown that for the existence of dark energy[8]. this simple time dependent background, the infrared (IR) From the theoretical point of view, past experience tell and ultraviolet (UV) behaviors are changed drastically. us that strings probe space time in a very different way In section III, we discuss the stability of the cosmological than we may intuitively think. A compelling example solutions. We find that the pseudotachyon modes do not is the emergence of M theory, where when we study the imply the instability of the background. In section IV, strong coupling behavior of Type IIA string theory, the we study the cosmological solutions with the compact theory develops a new dimension, and the theory is then hyperbolic spatial section in critical dimension and we formulated in 11 dimensions! The extended nature of the find that the theory is indeed supercritical. In section V, string reveals many amazing things about the property of the relation of this background with supercritical string space time (T duality is one of the famous example). We cosmology is pointed out. 2 II. COSMOLOGICAL SOLUTIONS AND THE non-trivial; the remarkable thing is that this background QUANTIZATION with the compact hyperbolic section is indeed supercrit- ical. Before we get into the discussion of the hyperbolic To study the nontrivial space time background, it is case, it is very useful to give a review of the well-studied quite useful to start with the string world sheet non- solutions with κ = 0 and κ = 1, we will find some generic linear sigma model in curved space time. The bosonic features of time-dependent solutions. string action reads: κ 1 2 αβ µν A. Spectrum of Cosmological Solutions with = 0 = ′ d σ√ h[h ∂ X ∂ X G + S −4πα − α µ β ν and κ = 1 Z P ′ The simplest time dependent background is the so- iǫ Bµν ∂ X ∂ X + α RΦ(X)]. (1) αβ α µ β ν called supercritical linear dilaton background (SCLD), The equation of motion for the background fields can this solution can be derived if we take κ = 0 in our sec- be derived from the requirement that the β functions ond class of solutions. A discussion of this background of this two dimensional field theory vanish to one-loop may also be found in[11]. Regardless of its simplicity, we order, where the β functions can be found in [9][10]. can definitely learn a lot from studying this background We are interested in finding the four dimensional FRW due to the nontrivial time dependence of the solution. type solution (we assume that the four dimensional CFT The background reads: is decoupled from the internal Conformal Field Theory 0 Gµν = ηµν , Φ= 2QX . (6) (CFT)). We have found in [6] the following simple and − asymptotically unique cosmological solutions (we only The central charge of this CFT is write the four large dimension with the internal part of c = D 12Q2. (7) the background suppressed) : − 1. The Einstein static universe, with the background The anomaly cancelation requires that c = 26 for bosonic fields as (the metric is expressed in the Einstein frame, string, so for Q real, it is clear that the space time di- the relation of the metric between string frame and Ein- mension is larger than the critical dimension. stein frame is gE = e 2Φ/(D 2)GS ) µν − − µν The effect of the linear dilaton is to change the world Φ0 0 sheet energy-momentum tensor to the form (where we Φ=Φ0, b =2e− √κX . (2) ′ have set α = 2) Here we have a constant dilaton, while the axion b is re- 1 µ 2 0 lated to the NS field strength through the duality relation Tzz = ∂zX ∂zXµ Q∂z X . (8) 2φ ρ −2 − Hλµν = e ǫλµνρ b, and the metric is given by ▽ We can quantize the theory by using the conventional dr2 (ds)2 = (dX0)2 + [ + r2(dθ2 + sin2 θdφ2)]. mode expansion, whic is possible, because the theory is − 1 κr2 Weyl invariant and reparameterization invariant, and be- − (3) cause we can gauge fix the world-sheet metric as hij = 2. The second class of solutions involves both a non- δij ). trivial dilaton and the non-trivial axion field. The solu- The Virasoro generators are [6] tion is 0 1 µ 0 QX Ln =: an kakµ :+iQ(n + 1)an, (9) 0 2 e 2 − Φ= 2QX , b =2Q √κ( ), (4) k − Q X where we consider the left moving modes only; the right while the metric is given by moving modes have the similar form. The commutation relation for the creation and annihilation operators are dr2 (ds)2 = (dt)2 + t2[ + r2(dθ2 + sin2 θdφ2)]. (5) familiar: − 1 κr2 − [aµ ,aν ]= mηµν δ . (10) The κ is the curvature of the spatial slice, and X0 is the m n m+n world sheet time coordinate, it is related to the Einstein The hermiticity relations of the Virasoro generators 0 1 QX + time coordinate as t = e . The central charge deficit Ln = L n dictate that Q − is δc = cI 22 (1 + κ), so only for κ = 1, can we find µ+ µ µ0 − ∝ − an = a n + i2Qη δn. (11) the cosmological solutions in critical dimension, while for − the flat and positive curvature the theory is supercritical. This relation means that the zeroth component has a However, for κ = 1, the solution is trivial due to the fixed imaginary part: fact that this background− is only part of the flat space. We can take a quotient of this space to make the solution p0 = E + iQ.
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