  It is shown by three-dimensional QED particle-in-cell simulation that as a
laser pulse of 2.5 PW and 20 fs propagates along a sub-wavelength-wide solid
wire, directional synchrotron $\gamma-$rays along the wire surface can be
efficiently generated. With 8\% energy conversion from the pulse, the
$\gamma-$rays contains $10^{12}$ photons between 5 and 500 MeV within 10 fs
duration, corresponding to peak brilliance of $10^{27}$ photons ${\rm s^{-1}~
mrad^{-2}~ mm^{-2}}$ per 0.1\% bandwidth. The brilliance and photon energy are
respectively 2 and 3 orders of magnitude higher than the highest values of
synchrotron radiation facilities. The radiation is attributed to the generation
of nC, GeV electron beams well guided along the wire surface and their wiggling
motion in strong electrostatic and magnetostatic fields induced at the
high-density-wire surface. In particular, these quasistatic fields are so
strong that QED effects already play a significant role for the $\gamma-$ray
radiation. With the laser power $P_0$ ranging from 0.5 PW to 5 PW available
currently, this scheme can robustly produce $\gamma-$rays peaked at $1^\circ$
with few-mrad divergence and the photon energy and number roughly scales with
$P_0$ and $P_0^{3/2}$, respectively. Our scheme embraces both the merits of
high directionality comparable to those based upon laser wakefield acceleration
and high charge comparable to those based upon laser-solid interaction.
