  We conduct a deep narrow-band imaging survey of 13 Ly$\alpha$ blobs (LABs)
located in the SSA22 proto-cluster at z~3.1 in the CIV and HeII emission lines
in an effort to constrain the physical process powering the Ly$\alpha$ emission
in LABs. Our observations probe down to unprecedented surface brightness limits
of 2.1 $-$ 3.4 $\times$ 10$^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ per 1
arcsec$^2$ aperture (5$\sigma$) for the HeII$\lambda$1640 and CIV$\lambda$1549
lines, respectively. We do not detect extended HeII and CIV emission in any of
the LABs, placing strong upper limits on the HeII/Ly$\alpha$ and CIV/Ly$\alpha$
line ratios, of 0.11 and 0.16, for the brightest two LABs in the field. We
conduct detailed photoionization modeling of the expected line ratios and find
that, although our data constitute the deepest ever observations of these
lines, they are still not deep enough to rule out a scenario where the
Ly$\alpha$ emission is powered by the ionizing luminosity of an obscured AGN.
Our models can accommodate HeII/Ly$\alpha$ and CIV/Ly$\alpha$ ratios as low as
$\simeq$0.05 and $\simeq$0.07 respectively, implying that one needs to reach
surface brightness as low as 1 $-$ 1.5 $\times$ 10$^{-18}$ erg s$^{-1}$
cm$^{-2}$ arcsec$^{-2}$ (at 5$\sigma$) in order to rule out a photoionization
scenario. These depths will be achievable with the new generation of
image-slicing integral field units such as VLT/MUSE or Keck/KCWI. We also model
the expected HeII/Ly$\alpha$ and CIV/Ly$\alpha$ in a different scenario, where
Ly$\alpha$ emission is powered by shocks generated in a large-scale superwind,
but find that our observational constraints can only be met for shock
velocities $v_{\rm s} \gtrsim$ 250 km s$^{-1}$, which appear to be in conflict
with recent observations of quiescent kinematics in LABs.
