Free Energy and Critical Temperature in Eleven Dimensions

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 R × T , 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 Tcr ∼= 0.31 l−1 P (so it is very small in string units). January 2001 ∗ e-mail address: [email protected]