  This paper examines the question for global regularity for the Boussinesq
equation with critical fractional dissipation. The main result states that the
system admits global regular solutions for all (reasonably) smooth and decaying
data, as long as $\al>2/3$. This significantly improves upon some recent works.
The main new idea is the introduction of a new, second generation
Hmidi-Keraani-Rousset type, change of variables, which further improves the
linear derivative in temperature term in the vorticity equation. This approach
is then complemented by new set of commutator estimates, which may be of
independent interest.
