Mass Spectrum of Strings in Anti de Sitter Spacetime

We perform string quantization in anti de Sitter (AdS) spacetime. The string motion is stable, oscillatory in time with real frequencies $\omega_n= \sqrt{n^2+m^2\alpha'^2H^2}$ and the string size and energy are bounded. The string fluctuations around the center of mass are well behaved. We find the mass formula which is also well behaved in all regimes. There is an {\it infinite} number of states with arbitrarily high mass in AdS (in de Sitter (dS) there is a {\it finite} number of states only). The critical dimension at which the graviton appears is $D=25,$ as in de Sitter space. A cosmological constant $\Lambda\neq 0$ (whatever its sign) introduces a {\it fine structure} effect (splitting of levels) in the mass spectrum at all states beyond the graviton. The high mass spectrum changes drastically with respect to flat Minkowski spacetime. For $\Lambda\sim \mid\Lambda\mid N^2,$ {\it independent} of $\alpha',$ and the level spacing {\it grows} with the eigenvalue of the number operator, $N.$ The density of states $\rho(m)$ grows like \mbox{Exp}[(m/\sqrt{\mid\Lambda\mid}\;)^{1/2}] (instead of \rho(m)\sim\mbox{Exp}[m\sqrt{\alpha'}] as in Minkowski space), thus {\it discarding} the existence of a critical string temperature. For the sake of completeness, we also study the quantum strings in the black string background, where strings behave, in many respects, as in the ordinary black hole backgrounds. The mass spectrum is equal to the mass spectrum in flat Minkowski space.Comment: 31 pages, Latex, DEMIRM-Paris-9404

Semi-Classical Quantization of Circular Strings in De Sitter and Anti De Sitter Spacetimes

We compute the {\it exact} equation of state of circular strings in the (2+1) dimensional de Sitter (dS) and anti de Sitter (AdS) spacetimes, and analyze its properties for the different (oscillating, contracting and expanding) strings. The string equation of state has the perfect fluid form $P=(\gamma-1)E,$ with the pressure and energy expressed closely and completely in terms of elliptic functions, the instantaneous coefficient $\gamma$ depending on the elliptic modulus. We semi-classically quantize the oscillating circular strings. The string mass is $m=\sqrt{C}/(\pi H\alpha'),\;C$ being the Casimir operator, $C=-L_{\mu\nu}L^{\mu\nu},$ of the $O(3,1)$-dS [$O(2,2)$-AdS] group, and $H$ is the Hubble constant. We find \alpha'm^2_{\mbox{dS}}\approx 5.9n,\;(n\in N_0), and a {\it finite} number of states N_{\mbox{dS}}\approx 0.17/(H^2\alpha') in de Sitter spacetime; m^2_{\mbox{AdS}}\approx 4H^2n^2 (large $n\in N_0$) and N_{\mbox{AdS}}=\infty in anti de Sitter spacetime. The level spacing grows with $n$ in AdS spacetime, while is approximately constant (although larger than in Minkowski spacetime) in dS spacetime. The massive states in dS spacetime decay through tunnel effect and the semi-classical decay probability is computed. The semi-classical quantization of {\it exact} (circular) strings and the canonical quantization of generic string perturbations around the string center of mass strongly agree.Comment: Latex, 26 pages + 2 tables and 5 figures that can be obtained from the authors on request. DEMIRM-Obs de Paris-9404

String Driven Cosmology and its Predictions

We present a minimal model for the Universe evolution fully extracted from effective String Theory. This model is by its construction close to the standard cosmological evolution, and it is driven selfconsistently by the evolution of the string equation of state itself. The inflationary String Driven stage is able to reach enough inflation, describing a Big Bang like evolution for the metric. By linking this model to a minimal but well established observational information, (the transition times of the different cosmological epochs), we prove that it gives realistic predictions on early and current energy density and its results are compatible with General Relativity. Interestingly enough, the predicted current energy density is found Omega = 1 and a lower limit Omega \geq 4/9 is also found. The energy density at the exit of the inflationary stage also gives | Omega |_{inf}=1. This result shows an agreement with General Relativity (spatially flat metric gives critical energy density) within an inequivalent Non-Einstenian context (string low energy effective equations). The order of magnitude of the energy density-dilaton coupled term at the beginning of the radiation dominated stage agrees with the GUT scale. The predicted graviton spectrum is computed and analyzed without any free parameters. Peaks and asymptotic behaviours of the spectrum are a direct consequence of the dilaton involved and not only of the scale factor evolution. Drastic changes are found at high frequencies: the dilaton produces an increasing spectrum (in no string cosmologies the spectrum is decreasing). Without solving the known problems about higher order corrections and graceful exit of inflation, we find this model closer to the observational Universe than the current available string cosmology scenarii.Comment: LaTex, 22 pages, Lectures delivered at the Chalonge School, Nato ASI: Phase Transitions in the Early Universe: Theory and Observations. To appear in the Proceedings, Editors H. J. de Vega, I. Khalatnikov, N. Sanchez. (Kluwer Pub

String dynamics in cosmological and black hole backgrounds: The null string expansion

We study the classical dynamics of a bosonic string in the $D$--dimensional flat Friedmann--Robertson--Walker and Schwarzschild backgrounds. We make a perturbative development in the string coordinates around a {\it null} string configuration; the background geometry is taken into account exactly. In the cosmological case we uncouple and solve the first order fluctuations; the string time evolution with the conformal gauge world-sheet $\tau$--coordinate is given by $X^0(\sigma, \tau)=q(\sigma)\tau^{1\over1+2\beta}+c^2B^0(\sigma, \tau)+\cdots$, $B^0(\sigma,\tau)=\sum_k b_k(\sigma)\tau^k$ where $b_k(\sigma)$ are given by Eqs.\ (3.15), and $\beta$ is the exponent of the conformal factor in the Friedmann--Robertson--Walker metric, i.e. $R\sim\eta^\beta$. The string proper size, at first order in the fluctuations, grows like the conformal factor $R(\eta)$ and the string energy--momentum tensor corresponds to that of a null fluid. For a string in the black hole background, we study the planar case, but keep the dimensionality of the spacetime $D$ generic. In the null string expansion, the radial, azimuthal, and time coordinates $(r,\phi,t)$ are $r=\sum_n A^1_{n}(\sigma)(-\tau)^{2n/(D+1)}~,$ $\phi=\sum_n A^3_{n}(\sigma)(-\tau)^{(D-5+2n)/(D+1)}~,$ and $t=\sum_n A^0_{n} (\sigma)(-\tau)^{1+2n(D-3)/(D+1)}~.$ The first terms of the series represent a {\it generic} approach to the Schwarzschild singularity at $r=0$. First and higher order string perturbations contribute with higher powers of $\tau$. The integrated string energy-momentum tensor corresponds to that of a null fluid in $D-1$ dimensions. As the string approaches the $r=0$ singularity its proper size grows indefinitely like $\sim(-\tau)^{-(D-3)/(D+1)}$. We end the paper giving three particular exact string solutions inside the black hole.Comment: 17 pages, REVTEX, no figure

Strings in Cosmological and Black Hole Backgrounds: Ring Solutions

The string equations of motion and constraints are solved for a ring shaped Ansatz in cosmological and black hole spacetimes. In FRW universes with arbitrary power behavior [R(X^0) = a\;|X^0|^{\a}\, ], the asymptotic form of the solution is found for both $X^0 \to 0$ and $X^0 \to \infty$ and we plot the numerical solution for all times. Right after the big bang ($X^0 = 0$), the string energy decreasess as $R(X^0)^{-1}$ and the string size grows as $R(X^0)$ for $0 1$. Very soon [ $X^0 \sim 1$] , the ring reaches its oscillatory regime with frequency equal to the winding and constant size and energy. This picture holds for all values of \a including string vacua (for which, asymptotically, \a = 1). In addition, an exact non-oscillatory ring solution is found. For black hole spacetimes (Schwarzschild, Reissner-Nordstr\oo m and stringy), we solve for ring strings moving towards the center. Depending on their initial conditions (essentially the oscillation phase), they are are absorbed or not by Schwarzschild black holes. The phenomenon of particle transmutation is explicitly observed (for rings not swallowed by the hole). An effective horizon is noticed for the rings. Exact and explicit ring solutions inside the horizon(s) are found. They may be interpreted as strings propagating between the different universes described by the full black hole manifold.Comment: Paris preprint PAR-LPTHE-93/43. Uses phyzzx. Includes figures. Text and figures compressed using uufile

A method for solve integrable A2A_2 spin chains combining different representations

A non homogeneous spin chain in the representations $\{3 \}$ and $\{3^*\}$ of $A_2$ is analyzed. We find that the naive nested Bethe ansatz is not applicable to this case. A method inspired in the nested Bethe ansatz, that can be applied to more general cases, is developed for that chain. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations different from that for a homogeneous chain. A conjecture about the form of the solutions for more general chains is presented. PACS: 75.10.Jm, 05.50+q 02.20 SvComment: PlainTeX, harvmac, 13 pages, 3 figures, to appear in Phys. Rev.

QFT, String Temperature and the String Phase of De Sitter Space-time

The density of mass levels \rho(m) and the critical temperature for strings in de Sitter space-time are found. QFT and string theory in de Sitter space are compared. A `Dual'-transform is introduced which relates classical to quantum string lengths, and more generally, QFT and string domains. Interestingly, the string temperature in De Sitter space turns out to be the Dual transform of the QFT-Hawking-Gibbons temperature. The back reaction problem for strings in de Sitter space is addressed selfconsistently in the framework of the `string analogue' model (or thermodynamical approach), which is well suited to combine QFT and string study.We find de Sitter space-time is a self-consistent solution of the semiclassical Einstein equations in this framework. Two branches for the scalar curvature R(\pm) show up: a classical, low curvature solution (-), and a quantum high curvature solution (+), enterely sustained by the strings. There is a maximal value for the curvature R_{\max} due to the string back reaction. Interestingly, our Dual relation manifests itself in the back reaction solutions: the (-) branch is a classical phase for the geometry with intrinsic temperature given by the QFT-Hawking-Gibbons temperature.The (+) is a stringy phase for the geometry with temperature given by the intrinsic string de Sitter temperature. 2 + 1 dimensions are considered, but conclusions hold generically in D dimensions.Comment: LaTex, 24 pages, no figure

Multi-String Solutions by Soliton Methods in De Sitter Spacetime

{\bf Exact} solutions of the string equations of motion and constraints are {\bf systematically} constructed in de Sitter spacetime using the dressing method of soliton theory. The string dynamics in de Sitter spacetime is integrable due to the associated linear system. We start from an exact string solution $q_{(0)}$ and the associated solution of the linear system $\Psi^{(0)} (\lambda)$, and we construct a new solution $\Psi(\lambda)$ differing from $\Psi^{(0)}(\lambda)$ by a rational matrix in $\lambda$ with at least four poles $\lambda_{0},1/\lambda_{0},\lambda_{0}^*,1/\lambda_{0}^*$. The periodi- city condition for closed strings restrict $\lambda _{0}$ to discrete values expressed in terms of Pythagorean numbers. Here we explicitly construct solu- tions depending on $(2+1)$-spacetime coordinates, two arbitrary complex numbers (the 'polarization vector') and two integers $(n,m)$ which determine the string windings in the space. The solutions are depicted in the hyperboloid coor- dinates $q$ and in comoving coordinates with the cosmic time $T$. Despite of the fact that we have a single world sheet, our solutions describe {\bf multi- ple}(here five) different and independent strings; the world sheet time $\tau$ turns to be a multivalued function of $T$.(This has no analogue in flat space- time).One string is stable (its proper size tends to a constant for $T\to\infty$, and its comoving size contracts); the other strings are unstable (their proper sizes blow up for $T\to\infty$, while their comoving sizes tend to cons- tants). These solutions (even the stable strings) do not oscillate in time. The interpretation of these solutions and their dynamics in terms of the sinh- Gordon model is particularly enlighting.Comment: 25 pages, latex. LPTHE 93-44. Figures available from the auhors under reques

Planetoid strings : solutions and perturbations

A novel ansatz for solving the string equations of motion and constraints in generic curved backgrounds, namely the planetoid ansatz, was proposed recently by some authors. We construct several specific examples of planetoid strings in curved backgrounds which include Lorentzian wormholes, spherical Rindler spacetime and the 2+1 dimensional black hole. A semiclassical quantisation is performed and the Regge relations for the planetoids are obtained. The general equations for the study of small perturbations about these solutions are written down using the standard, manifestly covariant formalism. Applications to special cases such as those of planetoid strings in Minkowski and spherical Rindler spacetimes are also presented.Comment: 24 pages (including two figures), RevTex, expanded and figures adde

Integrable Generalized Thirring Model

We derive the conditions that the coupling constants of the Generalized Thirring Model have to satisfy in order for the model to admit an infinite number of commuting classical conserved quantities. Our treatment uses the bosonized version of the model, with periodic boundary conditions imposed on the space coordinate. Some explicit examples that satisfy these conditions are discussed. We show that, with a different set of boundary conditions, there exist additional conserved quantities, and we find the Poisson Bracket algebra satisfied by them.Comment: Final version to be published in Nucl.Phys.B. Only minor changes. An equation was deleted and a conclusion revised, and a few comments were added. Harvmac, 15 pages. Full postscript available from http://theor1.lbl.gov/www/theorgroup/papers/39040.p