Strings propagating in the (2+1)-dimensional black hole anti-de Sitter spacetime.

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We study the string propagation in the 2+1 black hole anti--de Sitter (2+1 BHAdS) background. 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 nonrotating black hole backgrounds the circular strings have a maximal bounded size ${\mathit{r}}_{\mathit{m}}$; they contract and collapse into r=0. No indefinitely growing strings nor multistring solutions are present in these backgrounds. In rotating spacetimes, both the 2+1 BHAdS and the ordinary Kerr-AdS backgrounds, 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 BHAdS spacetime), in which expanding unbounded strings and multistring solutions appear.