This paper examines the global regularity problem on the two-dimensional
incompressible Boussinesq equations with fractional dissipation, given by
$\Lambda^\alpha u$ in the velocity equation and by $\Lambda^\beta \theta$ in
the temperature equation, where $\Lambda=\sqrt{-\Delta}$ denotes the Zygmund
operator. We establish the global existence and smoothness of classical
solutions when $(\alpha,\beta)$ is in the critical range:
$\alpha>\frac{\sqrt{1777}-23}{24} =0.798103..$, $\beta>0$ and $\alpha+ \beta
=1$. This result improves the previous work of Jiu, Miao, Wu and Zhang
\cite{JMWZ} which obtained the global regularity for $\alpha>
\frac{23-\sqrt{145}}{12} \approx 0.9132$, $\beta>0$ and $\alpha+ \beta =1$.
