Free energy and critical temperature in eleven dimensions

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We compute the one-loop contribution to the free energy in eleven-dimensional supergravity, with the eleventh dimension compactified on a circle of radius R₁₁. We find a finite result, which, in a small radius expansion, has the form of the type IIA supergravity free energy plus non-perturbative corrections in the string coupling gₐ, whose coefficients we determine. We then study type IIA superstring theory at finite temperature in the strong coupling regime by considering M-theory on R⁹× T², one of the sides of the torus being the euclidean time direction, where fermions obey antiperiodic boundary conditions. We find that a certain winding membrane state becomes tachyonic above some critical temperature, which depends on gₐ. At weak coupling, it coincides with the Hagedorn temperature, at large coupling it becomes T cr 0.31 lₚ⁻¹ (so it is very small in string units).