$AdS_3$ soliton of Witten - for Hawking-Page transition

Question

Are there explicit AdS$_3$ soliton solution?

in the sense of Witten's Anti De Sitter Space And Holography and Hawking-Page transition paper, by doing a $$\tau_E, y ,r \to y, \tau_E ,r$$

from a geometry with a Euclidean time $\tau_E$ cigar space and a compactified $y$ circle

to a geometry with a $y$ cigar space and a compactified Euclidean time $\tau_E$ circle

If yes, with a AdS3 soliton solution, what are the form of solutions?

If not, without a AdS3 soliton solution, what are the obstructions?

Answer 1

There is an explicitly three-dimensional version of a Hawking-Page transition in AdS space. It is given by a transition between an asymptotically $AdS_3$ spacetime and the BTZ (Bañados-Teitelboim-Zanelli) black hole. The line element of thermal (euclidean) $AdS_3$ space is given by

$$ds^2=\left(1+r^2/L^2\right)dt^2+\left(1+r^2/L^2\right)^{-1}dr^2+r^2d\phi^2,$$

where the periodicity of $t$ is given by the inverse temperature and $L$ is the $AdS$ radius. At a temperature that is determined by the point when the compact circles of both geometries match, the system undergoes a phase transition to the BTZ black hole geometry given by

$$ds^2=\left(-M+r^2/L^2\right)dt^2+\left(-M+r^2/L^2\right)^{-1}dr^2+r^2d\phi^2,$$

with $M$ as the mass of the black hole.