Closed String Thermal Torus From Thermo Field Dynamics
  In this Letter a topological interpretation for the string thermal vacuum in the Thermo Field Dynamics (TFD) approach is given. As a consequence, the relationship between the Imaginary Time and TFD formalisms is achieved when both are used to study closed strings at finite temperature. The TFD approach starts by duplicating the system's degrees of freedom, defining an auxiliary (tilde) string. In order to lead the system to finite temperature a Bogoliubov transformation is implemented. We show that the effect of this transformation is to glue together the string and the tilde string to obtain a torus. The thermal vacuum appears as the boundary state for this identification. Also, from the thermal state condition, a Kubo-Martin-Schwinger condition for the torus topology is derived. 
M. C. B Abdalla, A. L. Gadelha and Daniel L. Nedel