We compute the 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 = (γ − 1)E, with the pressure and energy expressed closely and completely in terms of elliptic functions, the instantaneous coefficient γ depending on the elliptic modulus. We semi-classically quantize the oscillating circular strings. The string AdS ≈ 4H2 n 2 (large n ∈ N 0 ) and N AdS = ∞ 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 semiclassical quantization of exact (circular) strings and the canonical quantization of generic string perturbations around the string center of mass strongly agree.

The exact general evolution of circular strings in 2 + 1 dimensional de Sitter spacetime is described closely and completely in terms of elliptic functions. The evolution depends on a constant parameter b, related to the string energy, and falls into three classes depending on whether b < 1/4 (oscillatory motion), b = 1/4 (degenerated, hyperbolic motion) or b > 1/4 (unbounded motion). The novel feature here is that one single world-sheet generically describes infinitely many (different and independent) strings. The world-sheet time τ is an infinite-valued function of the string physical time, each branch yields a different string. This has no analogue in flat spacetime. We compute the string energy E as a function of the string proper size S, and analyze it for the expanding and oscillating strings. For expanding strings (Ṡ > 0): E = 0 even at S = 0, E decreases for small S and increases ∝ S for large S. For an oscillating string (0 ≤ S ≤ S max ), the average energy < E > over one oscillation period is expressed as a function of S max as a complete elliptic integral of the third kind.For each b, the two independent solutions S + and S − are analyzed. For b < 1/4, all the strings of the S − solution are unstable (S max = ∞) and never collapse to a point (S min = 0). S + describes one stable (S max is bounded) oscillating string and < E > is an increasing function of b for 0 ≤ b ≤ 1/4. For b > 1/4, all strings (for both S + and S − ) are unstable and have a collapse during their evolution. For b = 1/4, S − describes two strings (one stable and one unstable for large de Sitter radius), while S + describes one stable nonoscillating string.2

The static cylindrically symmetric solutions of the gravitating Abelian Higgs model form a two parameter family. In this paper we give a complete classification of the string-like solutions of this system. We show that the parameter plane is composed of two different regions with the following characteristics: One region contains the standard asymptotically conic cosmic string solutions together with a second kind of solutions with Melvin-like asymptotic behavior. The other region contains two types of solutions with bounded radial extension. The border between the two regions is the curve of maximal angular deficit of 2π.

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.