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It is shown by three-dimensional QED particle-in-cell simulation that as alaser pulse of 2.5 PW and 20 fs propagates along a sub-wavelength-wide solidwire, directional synchrotron $\gamma-$rays along the wire surface can beefficiently generated. With 8\% energy conversion from the pulse, the$\gamma-$rays contains $10^{12}$ photons between 5 and 500 MeV within 10 fsduration, corresponding to peak brilliance of $10^{27}$ photons ${\rm s^{-1}~mrad^{-2}~ mm^{-2}}$ per 0.1\% bandwidth. The brilliance and photon energy arerespectively 2 and 3 orders of magnitude higher than the highest values ofsynchrotron radiation facilities. The radiation is attributed to the generationof nC, GeV electron beams well guided along the wire surface and their wigglingmotion in strong electrostatic and magnetostatic fields induced at thehigh-density-wire surface. In particular, these quasistatic fields are sostrong that QED effects already play a significant role for the $\gamma-$rayradiation. With the laser power $P_0$ ranging from 0.5 PW to 5 PW availablecurrently, 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 ofhigh directionality comparable to those based upon laser wakefield accelerationand high charge comparable to those based upon laser-solid interaction.

Collimated Ultra-Bright Gamma-Rays from a PW-Laser-Driven Wire Wiggler