Thermal $D$-Brane Boundary States from Green-Schwarz Superstrings

In this paper we thermalize the type II superstrings in the GS formulation by applying the TFD formalism. The thermal boundary conditions on the thermal Hilbert space are obtained from the BPS $D$-brane boundary conditions at zero temperature. We show that thermal boundary states can be obtained by thermalization from the BPS $D$-branes at zero temperature. These new states can be interpreted as thermal $D$-branes. Next, we discuss the supersymmetry breaking of the thermal string in the TFD approach. We identify the broken supersymmetry with the $\epsilon$-transformation while the $\eta$-transformation is preserved. Also, we compute the thermal partition function and the entropy of the thermal string.