String Scale Thermal Anti-de Sitter Spaces
Résumé
We study finite temperature string scale AdS$_{3}$ backgrounds. One background is AdS$_{3}$× S$^{1}$× T$^{2}$ in which the anti-de Sitter space-time and the circle are at the radius $ \sqrt{\alpha^{\prime }} $. Using path integral techniques, we show that the bulk spectrum includes a continuum of states as well as Ramond-Ramond ground states that agree with those of the symmetric orbifold of the two-torus after second quantization. We also examine the one-loop free energy of the background AdS$_{3}$× S$^{1}$ at curvature radius $ \sqrt{2{\alpha}^{\prime }/3} $. In the space-time NSNS sector, the string theory spontaneously breaks conformal symmetry as well as R-charge conjugation symmetry. We prove that the minimum in the boundary energy is reached for a singly wound string. In the RR sector, we classify the infinite set of ground states with fractional R-charges. Moreover, we remark on the behaviour of critical temperatures as the curvature scale becomes smaller than the string scale.In an appendix, we derive the Hawking-Page transition in string theory by integrating a world sheet one-point function.