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

Exact solution of the SUq(n)SU_{q}(n) invariant quantum spin chains

The Nested Bethe Ansatz is generalized to open boundary conditions. This is used to find the exact eigenvectors and eigenvalues of the $A_{n-1}$ vertex model with fixed open boundary conditions and the corresponding $SU_{q}(n)$ invariant hamiltonian. The Bethe Ansatz equations obtained are solved in the thermodynamic limit giving the vertex model free energy and the hamiltonian ground state energy including the corresponding boundary contributions.Comment: 29 page

Exact String Solutions in 2+1-Dimensional De Sitter Spacetime

Exact and explicit string solutions in de Sitter spacetime are found. (Here, the string equations reduce to a sinh-Gordon model). A new feature without flat spacetime analogy appears: starting with a single world-sheet, several (here two) strings emerge. One string is stable and the other (unstable) grows as the universe grows. Their invariant size and energy either grow as the expansion factor or tend to constant. Moreover, strings can expand (contract) for large (small) universe radius with a different rate than the universe.Comment: 11 pages, Phyzzx macropackage used, PAR-LPTHE 92/32. Revised version with a new understanding of the previous result

Planetoid String Solutions in 3 + 1 Axisymmetric Spacetimes

The string propagation equations in axisymmetric spacetimes are exactly solved by quadratures for a planetoid Ansatz. This is a straight non-oscillating string, radially disposed, which rotates uniformly around the symmetry axis of the spacetime. In Schwarzschild black holes, the string stays outside the horizon pointing towards the origin. In de Sitter spacetime the planetoid rotates around its center. We quantize semiclassically these solutions and analyze the spin/(mass$^2$) (Regge) relation for the planetoids, which turns out to be non-linear.Comment: Latex file, 14 pages, two figures in .ps files available from the author

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

Exact String Solutions in Nontrivial Backgrounds

We show how the classical string dynamics in $D$-dimensional gravity background can be reduced to the dynamics of a massless particle constrained on a certain surface whenever there exists at least one Killing vector for the background metric. We obtain a number of sufficient conditions, which ensure the existence of exact solutions to the equations of motion and constraints. These results are extended to include the Kalb-Ramond background. The $D1$-brane dynamics is also analyzed and exact solutions are found. Finally, we illustrate our considerations with several examples in different dimensions. All this also applies to the tensionless strings.Comment: 22 pages, LaTeX, no figures; V2:Comments and references added; V3:Discussion on the properties of the obtained solutions extended, a reference and acknowledgment added; V4:The references renumbered, to appear in Phys Rev.

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

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

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

Strings Propagating in the 2+1 Dimensional Black Hole Anti de Sitter Spacetime

We study the string propagation in the 2+1 black hole anti de Sitter background (2+1 BH-ADS). We find the first and second order fluctuations around the string center of mass and obtain the expression for the string mass. The string motion is stable, all fluctuations oscillate with real frequencies and are bounded, even at $r=0.$ We compare with the string motion in the ordinary black hole anti de Sitter spacetime, and in the black string background, where string instabilities develop and the fluctuations blow up at $r=0.$ We find the exact general solution for the circular string motion in all these backgrounds, it is given closely and completely in terms of elliptic functions. For the non-rotating black hole backgrounds the circular strings have a maximal bounded size $r_m,$ they contract and collapse into $r=0.$ No indefinitely growing strings, neither multi-string solutions are present in these backgrounds. In rotating spacetimes, both the 2+1 BH-ADS and the ordinary Kerr-ADS, the presence of angular momentum prevents the string from collapsing into $r=0.$ The circular string motion is also completely solved in the black hole de Sitter spacetime and in the black string background (dual of the 2+1 BH-ADS spacetime), in which expanding unbounded strings and multi-string solutions appear.Comment: Latex, 54 pages + 2 tables and 4 figures (not included). PARIS-DEMIRM 94/01