Bosonic D-branes at finite temperature with an external field

TL;DR

This paper constructs bosonic boundary states at finite temperature with an external gauge field using thermo field dynamics, interpreting them as finite-temperature Dp-branes and analyzing their entropy.

Contribution

It introduces a method to formulate finite-temperature boundary states with gauge fields, extending the concept of Dp-branes to thermal settings.

Findings

Boundary states are constructed at finite temperature with external fields.

Two solutions to the boundary conditions are identified.

The entropy of the finite-temperature Dp-brane is computed and analyzed.

Abstract

Bosonic boundary states at finite temperature are constructed as solutions of boundary conditions at $T\neq 0$ for bosonic open strings with a constant gauge field $F_{ab}$ coupled to the boundary. The construction is done in the framework of thermo field dynamics where a thermal Bogoliubov transformation maps states and operators to finite temperature. Boundary states are given in terms of states from the direct product space between the Fock space of the closed string and another identical copy of it. By analogy with zero temperature, the boundary states heve the interpretation of $Dp$-brane at finite temperature. The boundary conditions admit two different solutions. The entropy of the closed string in a $Dp$-brane state is computed and analysed. It is interpreted as the entropy of the $Dp$-brane at finite temperature.