Thermal AdS(3), BTZ and competing winding modes condensation

We study the thermal physics of AdS(3) and the BTZ black hole when embedded in String theory. The exact calculation of the Hagedorn temperature in TAdS(3) is reinterpreted as the appearance of a winding tachyon both in AdS(3) and BTZ. We construct a dual framework for analyzing the phases of the system. In this dual framework, tachyon condensation and geometric capping appear on the same footing, bridging the usual gap of connecting tachyon condensation to modifications of geometry. This allows us to construct in a natural way a candidate for the unstable phase, analogous to a small black hole in higher dimensions. Additional peculiar effects associated with the Hagedorn temperature and the Hawking-Page transition, some to do with the asymptotic structure of AdS(3) and some with strong curvature effects, are analyzed and explained.