  The conductivity of the two-dimensional Hubbard model is particularly
relevant for high-temperature superconductors. Vertex corrections are expected
to be important because of strongly momentum dependent self-energies. We use
the Two-Particle Self-Consistent approach that satisfies crucial constraints
such as the Mermin-Wagner theorem, the Pauli principle and sum rules in order
to reach non-perturbative regimes. This approach is reliable from weak to
intermediate coupling. A functional derivative approach ensures that vertex
corrections are included in a way that satisfies the f sum-rule. The two types
of vertex corrections that we find are the antiferromagnetic analogs of the
Maki-Thompson and Aslamasov-Larkin contributions of superconducting
fluctuations to the conductivity but, contrary to the latter, they include
non-perturbative effects. The resulting analytical expressions must be
evaluated numerically. The calculations are impossible unless a number of
advanced numerical algorithms are used. A maximum entropy approach is specially
developed for analytical continuation of our results. The numerical results are
for nearest neighbor hoppings. In the pseudogap regime induced by
two-dimensional antiferromagnetic fluctuations, the effect of vertex
corrections is dramatic. Without vertex corrections the resistivity increases
as we enter the pseudogap regime. Adding vertex corrections leads to a drop in
resistivity, as observed in some high temperature superconductors. At high
temperature, the resistivity saturates at the Ioffe-Regel limit. At the quantum
critical point and beyond, the resistivity displays both linear and quadratic
temperature dependence and there is a correlation between the linear term and
the superconducting transition temperature. A hump is observed in the
mid-infrared range of the optical conductivity in the presence of
antiferromagnetic fluctuations.
