  We prove exponential decay of the off-diagonal correlation function in the
two-dimensional homogeneous Bose gas when a^2 \rho is small and the temperature
T satisfies T > 4 \pi \rho / \ln |\ln(a^2\rho). Here, a is the scattering
length of the repulsive interaction potential and \rho is the density. To
leading order in a^2 \rho, this bound agrees with the expected critical
temperature for superfluidity. In the three-dimensional Bose gas, exponential
decay is proved when \Delta T_c / T_c^0 > 5 \sqrt{a \rho^{1/3}}, where T_c^0 is
the critical temperature of the ideal gas. While this condition is not expected
to be sharp, it gives a rigorous upper bound on the critical temperature for
Bose-Einstein condensation.
