Quantum states, thermodynamic limits, and entropy in M theory

We discuss the matching of the BPS part of the spectrum for a (super) membrane, which gives the possibility of getting the membrane's results via string calculations. In the small coupling limit of M theory the entropy of the system coincides with the standard entropy of type IIB string theory (including the logarithmic correction term). The thermodynamic behavior at a large coupling constant is computed by considering M theory on a manifold with a topology T-2 x R-9. We argue that the finite temperature partition functions (brane Laurent series for p not equal 1) associated with the BPS p-brane spectrum can be analytically continued to well-defined functionals. It means that a finite temperature can be introduced in brane theory, which behaves like finite temperature field theory. In the limit p --> 0 (point particle limit) it gives rise to the standard behavior of thermodynamic quantities.